arXiv:0704.0542v2  [math.CO]  23 Apr 2007
Hilbert functions of points on
Schubert varieties in Orthogonal Grassmannians
K. N. Raghavan
Institute of Mathematical Sciences
C. I. T. Campus, Taramani
Chennai 600 113, INDIA
email: knr@imsc.res.in
Shyamashree Upadhyay
Chennai Mathematical Institute
Plot No. H1, SIPCOT IT Park
Padur Post, Siruseri 603 103, Tamilnadu, INDIA
email: shyama@cmi.ac.in
04 April 2007
Abstract
A solution is given to the following problem: how to compute the mul-
tiplicity, or more generally the Hilbert function, at a point on a Schubert
variety in an orthogonal Grassmannian. Standard monomial theory is
applied to translate the problem from geometry to combinatorics. The
solution of the resulting combinatorial problem forms the bulk of the pa-
per. This approach has been followed earlier to solve the same problem
for the Grassmannian and the symplectic Grassmannian.
As an application, we present an interpretation of the multiplicity as
the number of non-intersecting lattice paths of a certain kind.
Taking the Schubert variety to be of a special kind and the point to be
the “identity coset,” our problem specializes to a problem about Pfaffian
ideals treatments of which by different methods exist in the literature.
Also available in the literature is a geometric solution when the point is
a “generic singularity.”
Mathematics Subject Classification 2000: 05E15 (Primary), 13F50, 13P10, 14L35 (Sec-
ondary)
http://arxiv.org/abs/0704.0542v2
Contents
Introduction 4
Organization of the paper . . . . . . . . . . . . . . . . . . . . . . . . . 5
Important note added . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
I The theorem 7
1 The set up 7
1.1 The statement of the problem . . . . . . . . . . . . . . . . . . . . 7
1.2 Some convenient choices . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Reduction to the case n even . . . . . . . . . . . . . . . . . . . . 8
2 The theorem 9
2.1 Basic notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Two fundamental definitions . . . . . . . . . . . . . . . . . . . . 11
2.2.1 Definition of v-chain . . . . . . . . . . . . . . . . . . . . . 11
2.2.2 Definition of O-domination . . . . . . . . . . . . . . . . . . 11
2.3 The main theorem and its corollary . . . . . . . . . . . . . . . . . 11
II From geometry to combinatorics 13
3 Reduction to combinatorics 13
3.1 Homogeneous co-ordinate ring of the Schubert variety X(w) . . . 13
3.1.1 The line bundle L on Md(V ) . . . . . . . . . . . . . . . . . 13
3.1.2 The section qθ of L . . . . . . . . . . . . . . . . . . . . . . 13
3.1.3 Standard monomial theory for Md(V ) . . . . . . . . . . . . 14
3.2 Co-ordinate rings of affine patches and tangent cones of X(w) . . 14
3.2.1 Standard monomial theory for affine patches . . . . . . . . . 14
3.2.2 Standard monomial theory for tangent cones . . . . . . . . . 16
4 Further reductions 17
4.1 The main propositions . . . . . . . . . . . . . . . . . . . . . . . . 17
4.2 From the main propositions to the main theorem . . . . . . . . . 18
III The proof 21
5 Terminology and notation 21
5.1 Distinguished subsets . . . . . . . . . . . . . . . . . . . . . . . . . 21
5.1.1 Distinguished subsets of N . . . . . . . . . . . . . . . . . . 21
5.1.2 Attaching elements of I(d, 2d) to distinguished subsets of N 21
5.2 The involution # . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5.2.1 The involution # on I(d, 2d) . . . . . . . . . . . . . . . . . 21
5.2.2 The involution # on N and R . . . . . . . . . . . . . . . . 22
5.3 The subset SC attached to a v-chain C . . . . . . . . . . . . . . 22
5.3.1 Vertical and horizontal projections of an element of ON . . 22
5.3.2 The “connection” relation on elements of a v-chain . . . . . 22
5.3.3 The definition of SC . . . . . . . . . . . . . . . . . . . . . 23
5.3.4 The type of an element α of a v-chain C, and the set SC,α 23
6 O-depth 27
6.1 Definition of O-depth . . . . . . . . . . . . . . . . . . . . . . . . 27
6.2 O-depth and depth . . . . . . . . . . . . . . . . . . . . . . . . . . 29
6.3 O-depth and type . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
7 The map Oπ 34
7.1 Description of Oπ . . . . . . . . . . . . . . . . . . . . . . . . . . 34
7.2 Illustration by an example . . . . . . . . . . . . . . . . . . . . . . 35
7.3 A proposition about Sj,j+1 . . . . . . . . . . . . . . . . . . . . . 38
7.4 Proof of Proposition 7.1.1 . . . . . . . . . . . . . . . . . . . . . . 40
7.5 More observations . . . . . . . . . . . . . . . . . . . . . . . . . . 41
8 The map Oφ 42
8.1 Description of Oφ . . . . . . . . . . . . . . . . . . . . . . . . . . 42
8.2 Basic facts about Tw,j,j+1 and T
w,j,j+1 . . . . . . . . . . . . . . . 45
9 Some Lemmas 46
9.1 Lemmas from the Grassmannian case . . . . . . . . . . . . . . . . 46
9.2 Orthogonal analogues of Lemmas of 9.1 . . . . . . . . . . . . . . 49
9.3 Orthogonal analogues of some lemmas in [7] . . . . . . . . . . . . 51
9.4 More lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
10 The Proof 59
10.1 Proof of Proposition 4.1.1 . . . . . . . . . . . . . . . . . . . . . . 59
10.2 Proof that OφOπ = identity . . . . . . . . . . . . . . . . . . . . 62
10.3 Proof that OπOφ = identity . . . . . . . . . . . . . . . . . . . . 63
10.4 Proof of Proposition 4.1.3 . . . . . . . . . . . . . . . . . . . . . . 64
IV An Application 65
11 Multiplicity counts certain paths 65
11.1 Description and illustration . . . . . . . . . . . . . . . . . . . . . 65
11.2 Justification for the interpretation . . . . . . . . . . . . . . . . . 67
References 71
Index of definitions and notation 73
Introduction
In this paper the following problem is solved: given a Schubert variety in an or-
thogonal Grassmannian (by which is meant the variety of isotropic subspaces of
maximum possible dimension of a finite dimensional vector space with a sym-
metric non-degenerate form—see §1 for precise definitions) and an arbitrary
point on the Schubert variety, how to compute the multiplicity, or more gener-
ally the Hilbert function, of the local ring of germs of functions at that point.
In a sense, our solution is but a translation of the problem: we do not give
closed form formulas but alternative combinatorial descriptions. The meaning
of “alternative” will presently become clear.
The same problem for the Grassmannian was treated in [11, 8, 7, 9, 12]
and for the symplectic Grassmannian in [4]. The present paper is a sequel
to [11, 7, 9, 12, 4] and toes the same line as them. In particular, its strategy
is borrowed from them and runs as follows: first translate the problem from
geometry to combinatorics, or, more precisely, apply standard monomial theory
to obtain an initial combinatorial description of the Hilbert function (the earliest
version of the theory capable of handling Schubert varieties in an orthogonal
Grassmannian is to be found in [17]); then transform the initial combinatorial
description to obtain the desired alternative description. But that is easier said
than done.
While the problem makes sense for Schubert varieties of any kind and stan-
dard monomial theory itself is available in great generality [13, 15], the transla-
tion of the problem from geometry to combinatorics has been made—in [14]—
only for “minuscule1 generalized Grassmannians.” Orthogonal Grassmannians
being minuscule, this translation is available to us and we have an initial combi-
natorial description of the Hilbert function. As to the passage from the initial to
the alternative description—and this is where the content of the present paper
lies—neither the end nor the means is clear at the outset.
The first problem then is to find a good alternative description. But how to
measure the worth of an alternative description? The interpretation of multi-
plicity as the number of certain non-intersecting lattice paths (deduced in §11
from our alternative description) seems to testify to the correctness of our al-
ternative description, but we are not sure if there are others that are equally or
more correct.
The proof of the equivalence of the initial and alternative combinatorial de-
scriptions is, unfortunately, a little technically involved. It builds on the details
of the proofs of the corresponding equivalences in the cases of the Grassmannian
and the symplectic Grassmannian. In [10] it is shown that the equivalence in
the case of the Grassmannian is a kind of KRS correspondence, called “bounded
KRS.” The proof there is short and elegant and it would be nice to realise the
main result of the present paper too in a similar spirit as a kind of KRS corre-
spondence.
The initial description is in terms of “standard monomials” and the alterna-
1Symplectic Grassmannians are not minuscule but can be treated as if they were.
tive description in terms of “monomials in roots.” The equivalence of the two
descriptions thus gives a bijective correspondence between standard monomials
and monomials in roots. Roughly—but not actually—the correspondence maps
each standard monomial to its initial term (with respect to a certain monomial
order). Thus it is natural to wonder whether we can compute the initial ideal
of the ideal of the tangent cone to the Schubert variety at the given point. We
believe that this can be done but that it is far more involved and difficult than
the corresponding computation for Grassmannians and symplectic Grassmanni-
ans (the natural set of generators of the ideal of the tangent cone do not form a
Gröbner basis unlike in those cases). If all goes well, the computation will soon
appear [16].
Taking the Schubert variety to be of a special kind and the point to be the
“identity coset,” our problem specializes to a problem about Pfaffian ideals con-
sidered in [5, 2]. On the other side of the spectrum from the identity coset, so to
speak, lie the “generic singularities,” points that are generic in the complement
of the open orbit of the stabiliser of the Schubert variety. For these, a geometric
solution to the problem appears in [1].
Given that our solution of the problem is but a translation, it makes sense
to ask if one can extract more tangible information—closed form formulas for
example—from our alternative description. See the papers quoted in the previ-
ous paragraph and also [3] for some answers in the special cases they consider.
Organization of the paper
The table of contents indicates how the paper is organized. There is a brief
description at the beginning of every subdivision of the contents therein. An
index of definitions and notation is included, for it would otherwise be difficult
to find the meanings of certain words and symbols.
Important note added
The recent article [6] treats some of the questions addressed here and some that
could be addressed by using the main result proved here. It includes:
• an interpretation of the multiplicity similar to ours.
• a closed formula for the multiplicity (as a specialization of a factorial Schur
function), thereby answering the question we raised above.
• a formula for the restriction to the torus fixed point of the equivariant
cohomology class of a Schubert variety.
The approach in [6] is quite different from ours. In fact, it is the opposite of
ours in that it circumvents the lack of results about initial ideals of tangent
cones, while our prime motivation is to remedy the lack. The starting points
in the two approaches are also different: [6] takes off from certain results of
Kostant-Kumar and Arabia on equivariant cohomology, while our launchpad is
standard monomial theory.
The appearance of [6] notwithstanding, our approach is worthwhile, for,
quite apart from the difference in starting points, there is no way, as far as we
can tell, to the Hilbert function via the approach of [6], nor to the initial ideal,
both of which are interesting in their own right.
Part I
The theorem
Definitions are recalled, the problem formulated, and the theorem stated.
1 The set up
In this section, we state the problem to be addressed after recalling the neces-
sary basic definitions, make some choices that are convenient for studying the
problem, and see why it is enough to focus on a particular case of the problem.
1.1 The statement of the problem
Fix an algebraically closed field of characteristic not equal to 2. Fix a vector
space V of finite dimension n over this field and a non-degenerate symmetric
bilinear form 〈 , 〉 on V . Let d be the integer such that either n = 2d or n =
2d+ 1. A linear subspace of V is said to be isotropic if the form 〈 , 〉 vanishes
identically on it. It is elementary to see that an isotropic subspace of V has
dimension at most d and that every isotropic subspace is contained in one of
dimension d. Denote by Md(V )
′ the closed sub-variety of the Grassmannian
of d-dimensional subspaces consisting of the points corresponding to isotropic
subspaces.
The orthogonal group O(V ) of linear automorphisms of V preserving 〈 , 〉
acts transitively on Md(V )
′, for by Witt’s theorem an isometry between sub-
spaces can be lifted to one of the whole vector space. If n is odd the special
orthogonal group SO(V ) (consisting of form preserving linear automorphisms
with trivial determinant) itself acts transitively on Md(V )
′. If n is even the spe-
cial orthogonal group SO(V ) does not act transitively on Md(V )
′, and Md(V )
has two connected components. We define the orthogonal Grassmannian Md(V )
to be Md(V )
′ if n is odd and to be one of the two components of Md(V )
′ if n
is even.
The Schubert varieties of Md(V ) are defined to be the B-orbit closures
in Md(V ) (with canonical reduced scheme structure), where B is a Borel sub-
group of SO(V ). The choice of B is immaterial, for any two of them are con-
jugate. The question that is tackled in this paper is this: given a point on a
Schubert variety in Md(V ), how to compute the multiplicity (and more gen-
erally, the Hilbert function) of the Schubert variety at the given point? The
answers are contained in Theorem 2.3.1 and Corollary 2.3.2. But in order to
make sense of those statements, we need some preparation.
1.2 Some convenient choices
We now make some choices that are convenient for the study of Schubert vari-
eties. For k an integer such that 1 ≤ k ≤ n, set k∗ := n + 1 − k. Fix a basis
e1, . . . , en of V such that
〈ei, ek〉 =
1 if i = k∗
0 otherwise
The advantage of this choice is: the elements of SO(V ) for which each ek is an
eigenvector form a maximal torus, and the elements that are upper triangular
with respect to this basis form a Borel subgroup (a linear transformation is upper
triangular if for each k, 1 ≤ k ≤ n, the image of ek under the transformation
is a linear combination of e1, . . . , ek). We denote this maximal torus and this
Borel subgroup by T and B respectively. Our Schubert varieties will be orbit
closures of this particular Borel subgroup B.
The B-orbits of Md(V )
′ are naturally indexed by its T -fixed points: each
orbit contains one and only one such point. The T -fixed points are evidently of
the form 〈ei1 , . . . , eid〉, where 1 ≤ i1 < . . . < id ≤ n and for each k, 1 ≤ k ≤ d,
there does not exist j, 1 ≤ j ≤ d, such that i∗k = ij—in other words, for each
ℓ, 1 ≤ ℓ ≤ n, such that ℓ 6= ℓ∗, exactly one of ℓ and ℓ∗ appears in {i1, . . . , id};
in addition, if n is odd, then d+ 1 does not appear in {i1, . . . , id}. Denote the
set of such d-element subsets {i1 < . . . < id} by I
n. We thus have a bijective
correspondence between I ′n and the B-orbits of Md(V )
′. Each B-orbit being
irreducible and open in its closure, it follows that B-orbit closures are indexed
by the B-orbits. Thus I ′n is an indexing set for B-orbit closures in Md(V )
Suppose that n is even—it will be shown presently that it is enough to
consider this case. As already observed, Md(V )
′ has two connected components
on each of which SO(V ) acts transitively. The B-orbits belong to one or the
other component accordingly as the parity of the cardinality of the number of
entries bigger than d in the corresponding element of I ′n. We take Md(V ) to
be the component in which these cardinalities are even. We let In denote the
subset of I ′n consisting of elements for which this cardinality is even. Schubert
varieties in Md(V ) are thus indexed by elements of In.
1.3 Reduction to the case n even
We now argue that it is enough to consider the case n even. Suppose that
n is odd. Let ñ := n + 1 and Ṽ be a vector space of dimension ñ with a
non-degenerate symmetric form. Let ẽ1, . . . , ẽen be a basis of Ṽ as in 1.2. Put
e := ẽd+1 and f := ẽd+2. Take λ to be an element of the field such that
λ2 = 1/2. We can take V to be the subspace of Ṽ spanned by the vectors
ẽ1, . . . , ẽd, λe + λf, ẽd+3, . . . , ẽen, and a basis of V to be these vectors in that
order.
There is a natural map from Md+1(Ṽ )
′ to Md(V ): intersecting with V an
isotropic subspace of Ṽ of dimension d + 1 gives an isotropic subspace of V of
dimension d. This map is onto, for every isotropic subspace of Ṽ (and hence
of V ) is contained in an isotropic subspace of Ṽ of dimension d + 1. It is
also elementary to see that the map is two-to-one (essentially because in a two-
dimensional space with a non-degenerate symmetric form there are two isotropic
lines), and that the two points in any fiber lie one in each component (there
is clearly an element in O(Ṽ ) \ SO(Ṽ ) that moves one element of the fiber to
the other, and so if there was an element of SO(Ṽ ) that also moved one point
to the other, the isotropy at the point would not be contained in SO(Ṽ ), a
contradiction).
We therefore get a natural isomorphism between Md+1(Ṽ ) and Md(V ). We
will now show that the B̃-orbits in Md+1(Ṽ ) correspond under the isomorphism
to B-orbits of Md(V ) (we denote by T̃ and B̃ the maximal torus and Borel
subgroups of SO(Ṽ ) as in §1.2). It will then follow that Schubert varieties in
Md+1(Ṽ ) are isomorphic to those in Md(V ) and the purpose of this subsection
will be achieved.
The group SO(V ) can be realized as the subgroup of SO(Ṽ ) consisting of
the elements that fix e − f . The isomorphism Md+1(Ṽ ) ∼= Md(V ) above is
equivariant for SO(V ), and we have T̃ ∩ SO(V ) = T and B̃ ∩ SO(V ) = B. It
should now be clear that the preimages in Md+1(Ṽ ) of two elements in the same
B-orbit of Md(V ) are in the same B̃-orbit: an element of B that moves one to
the other considered as an element of B̃ moves also the preimage of the one to
that of the other.
On the other hand, the preimages of distinct T -fixed points are distinct
T̃ -fixed points, the corresponding map from I ′n to Ien being given as follows:
i = {i1 < . . . < id} 7→
{̃i1, . . . , ĩd, d+ 1} if i ∈ In
{̃i1, . . . , ĩd, d+ 2} if i ∈ I
n \ In
where
ĩk =
ik if ik ≤ d
ik + 1 if ik ≥ d+ 2
(Note that d + 1 never occurs as an entry in any element of I ′n and that the
elements ĩ1, . . . , ĩd, d + 1 (respectively ĩ1, . . . , ĩd, d + 2) are not in increasing
order except in the trivial case i = {1 < . . . < d}.) Given that each B-orbit
has a T -fixed point and that distinct T̃ -fixed points belong to distinct B̃-orbits,
this implies that the preimages of two elements in distinct B-orbits belong to
distinct B̃-orbits, and the proof is over. �
2 The theorem
The purpose of this section is to state the main theorem and its corollary. We
first set down some basic notation and two fundamental definitions needed in
order to state the theorem.
2.1 Basic notation
We keep the terminology and notations of §1.1, 1.2. As observed in §1.3, it is
enough to consider the case n even. So from now on let n = 2d. Recall that,
for an integer k, 1 ≤ k ≤ 2d, k∗ := 2d + 1 − k. As observed in §1.2, Schubert
varieties in Md(V ) are indexed by In.
Since d now determines n, we will henceforth write I(d) instead of In. In
other words, I(d) is the set of d-element of subsets of {1, . . . , 2d} such that
• for each k, 1 ≤ k ≤ 2d, the subset contains exactly one of k, k∗, and
• the number of elements in the subset that exceed d is even.
We write I(d, 2d) for the set of all d-element subsets of {1, . . . , 2d}. There is a
natural partial order on I(d, 2d) and so also on I(d): v = (v1 < . . . < vd) ≤ w =
(w1 < . . . < wd) if and only if v1 ≤ w1, . . . , vd ≤ wd.
Given v ∈ I(d), the corresponding T -fixed point in Md(V ) (namely, the
span of ev1 , . . . , evd) is denoted e
v. Given w ∈ I(d), the corresponding Schubert
variety in Md(V ) (which, by definition, is the closure of the B-orbit of the T -
fixed point ew with canonical reduced scheme structure) is denoted X(w). The
point ev belongs to X(w) if and only if v ≤ w in the partial order just defined.
Since, under the natural action of B on X(w), each point of X(w) is in the
B-orbit of a T -fixed point ev for some v such that v ≤ w, it is enough to focus
attention on such T -fixed points.
For the rest of this section an element v of I(d) will remain fixed.
We will be dealing extensively with ordered pairs (r, c), 1 ≤ r, c ≤ 2d, such
that r is not and c is an entry of v. Let R denote the set of all such ordered
pairs, and set
N := {(r, c) ∈ R | r > c}
OR := {(r, c) ∈ R | r < c∗}
ON := {(r, c) ∈ R | r > c, r < c∗}
= OR ∩N
d := {(r, c) ∈ R | r = c∗}
diagonal
boundary
(r, c)
(c∗, c)
(r, r∗)
The picture shows a drawing of R. We think of r and c in (r, c) as row
index and column index respectively. The columns are indexed from left to
right by the entries of v in ascending order, the rows from top to bottom by
the entries of {1, . . . , 2d} \ v in ascending order. The points of d are those on
the diagonal, the points of OR are those that are (strictly) above the diag-
onal, and the points of N are those that are to the South-West of the poly-
line captioned “boundary of N”—we draw the boundary so that points on
the boundary belong to N. The reader can readily verify that d = 13 and
v = (1, 2, 3, 4, 6, 7, 10, 11, 13, 15, 18, 19, 22) for the particular picture drawn. The
points of ON indicated by solid circles form a v-chain (see §2.2.1 below).
We will be consideringmonomials, also calledmultisets, in some of these sets.
A monomial, as usual, is a subset with each member being allowed a multiplicity
(taking values in the non-negative integers). The degree of a monomial has also
the usual sense: it is the sum of the multiplicities in the monomial over all
elements of the set. The intersection of a monomial in a set with a subset
of the set has also the natural meaning: it is a monomial in the subset, the
multiplicities being those in the original monomial.
We will refer to d as the diagonal.
2.2 Two fundamental definitions
2.2.1 Definition of v-chain
Given two elements (R,C) and (r, c) in ON, we write (R,C) > (r, c) if R > r
and C < c (note that these are strict inequalities). An ordered sequence α, β, . . .
of elements of ON is called a v-chain if α > β > . . . . A v-chain α1 > . . . > αℓ
has head α1, tail αℓ, and length ℓ.
2.2.2 Definition of O-domination
To a v-chain C : α1 > α2 > . . . inON there corresponds, as described in §5.3.3, a
subset SC ofN which, as observed in Proposition 5.3.5, is “distinguished” in the
sense of §5.1.1. To a distinguished subset of N there corresponds, as described
below in §5.1.2, an element of I(d, 2d). Following these correspondences through,
we get an element of I(d, 2d) attached to the v-chain C. Let w(C) denote this
element—sometimes we write wC . (All this makes sense even when C is empty—
w(C) will turn out to be v itself in that case.)
Furthermore, as will be obvious from its definition, the monomial SC is
“symmetric” in the sense of §5.2.2 and contains evenly many elements of the
diagonal d. Thus, by Proposition 5.2.1, the element w(C) of I(d, 2d) belongs
to I(d).
An element w of I(d) is said toO-dominate C if w ≥ w(C), or, equivalently—
and this is important for the proofs—if w dominates in the sense of [7] the mono-
mial SC (for the proof of the equivalence, see [7, Lemma 5.5]). An element w of
I(d) O-dominates a monomial S of ON (repsectively of OR) if it O-dominates
every v-chain in S (respectively in S ∩ON).
2.3 The main theorem and its corollary
Theorem 2.3.1 Fix a positive integer d and elements v ≤ w of I(d). Let V
be a vector space of dimension 2d with a symmetric non-degenerate bilinear
form (over a field of characteristic not 2). Let X(w) be the Schubert variety
corresponding to w in the orthogonal Grassmannian Md(V ), and e
v the torus
fixed point of X(w) corresponding to v. Let Rwv denote the associated graded
ring with respect to the unique maximal ideal of the local ring of germs at ev of
functions on X(w). Then, for any non-negative integer m, the dimension as a
vector space of the homogeneous piece of Rwv of degree m equals the cardinality
of the set Sw(v)(m) of monomials of degree m of OR that are O-dominated
by w.
The proof of this theorem occupies us for most of this paper. It is reduced
in §3, by an application of standard monomial theory, to combinatorics. The
resulting combinatorial problem is solved in §4–10. For now, let us note the
following immediate consequence:
Corollary 2.3.2 The multiplicity at the point ev of the Schubert variety X(w)
equals the number of monomials in ON of maximal cardinality that are square-
free and O-dominated by w.
Proof: The proof of Corollary 2.2 of [7] holds verbatim here too. �
Part II
From geometry to combinatorics
The problem is translated from geometry to combinatorics. The main combi-
natorial results are formulated.
3 Reduction to combinatorics
In this section we translate the problem from geometry to combinatorics. In §3.1
we recall from [17] the theorem that enables the translation. The translation
itself is done in 3.2 and follows [14].
3.1 Homogeneous co-ordinate ring of the Schubert vari-
ety X(w)
3.1.1 The line bundle L on Md(V )
Let Md(V ) ⊆ Gd(V ) →֒ P(∧
dV ) be the Plücker embedding (where Gd(V )
denotes the Grassmannian of all d-dimensional subspaces of V ). The pull-back
toMd(V ) of the line bundle O(1) on P(∧
dV ) is the square of the ample generator
of the Picard group ofMd(V ). Letting L denote the ample generator, we observe
that it is very ample and want to describe the homogeneous coordinate rings
of Md(V ) and its Schubert subvarieties in the embedding defined by L.
3.1.2 The section qθ of L
For θ in I(d, 2d), let pθ denote the corresponding Plücker coordinate. Consider
the affine patch A of P(∧dV ) given by pǫ = 1, where ǫ := (1, . . . , d). The
intersection A ∩Gd(V ) of this patch with the Grassmannian is an affine space.
Indeed the d-plane corresponding to an arbitrary point z of A ∩ Gd(V ) has a
basis consisting of column vectors of a matrix of the form
where I is the identity matrix and A an arbitrary matrix both of size d × d.
The association z 7→ A is bijective. The restriction of a Plücker coordinate pθ
to A∩Gd(V ) is given by the determinant of a submatrix of size d× d of M , the
entries of θ determining the rows to be chosen from M to form the submatrix.
As can be readily verified, a point z of A ∩ Gd(V ) represents an isotropic
subspace if and only if the corresponding matrix A = (aij) is skew-symmetric
with respect to the anti-diagonal : aij + aj∗i∗ = 0, where the columns and rows
of A are numbered 1, . . . , d and d+1, . . . , 2d respectively. For example, if d = 4,
then a matrix that is skew-symmetric with respect to the anti-diagonal looks
like this: 
−d −c −b 0
−g −f 0 b
−i 0 f c
0 i g d
Since the set of these matrices is connected and contains the point that is
spanned by e1, . . . , ed, it follows that A∩Gd(V ) does not intersect the other com-
ponent of Md(V )
′. In other words, pǫ vanishes everywhere on Md(V )
′ \Md(V ).
Now suppose that θ belongs to I(d). Computing pθ/pǫ as a function on the
affine patch pǫ 6= 0, we see that it is the determinant of a skew-symmetric matrix
of even size, and therefore a square. The square root, which is determined up
to sign, is called the Pfaffian. This suggests that pθ itself is a square: more
precisely that there exists a section qθ of the line bundle L on Md(V ) such that
q2θ = pθ. A weight calculation confirms this to be the case. The qθ are also
called Pfaffians.
3.1.3 Standard monomial theory for Md(V )
A standard monomial in I(d) is a totally ordered sequence θ1 ≥ . . . ≥ θt (with
repetitions allowed) of elements of I(d). Such a standard monomial is said to
be w-dominated for w ∈ I(d) if w ≥ θ1. To a standard monomial θ1 ≥ . . . ≥ θt
in I(d) we associate the product qθ1 · · · qθt , where the qθ are the sections defined
above of the line bundle L. Such a product is also called a standard monomial
and it is said to be dominated by w for w ∈ I(d) if the underlying monomial in
I(d) is dominated by w. Standard monomial theory for Md(V ) says:
Theorem 3.1.1 (Seshadri [17]) Standard monomials qθ1 · · · qθr of degree r form
a basis for the space of forms of degree r in the homogeneous coordinate ring of
Md(V ) in the embedding defined by the ample generator L of the Picard group.
More generally, for w ∈ I(d), the w-dominated standard monomials of degree r
form a basis for the space of forms of degree r in the homogeneous coordinate
ring of the Schubert subvariety X(w) of Md(V ).
3.2 Co-ordinate rings of affine patches and tangent cones
of X(w)
From Theorem 3.1.1 one can deduce rather easily, as we now show, bases for
co-ordinate rings of affine patches of the form qv 6= 0 and of tangent cones of
Schubert varieties. An element v of I(d) will remain fixed for the rest of this
section. To simplify notation we will suppress explicit reference to v.
3.2.1 Standard monomial theory for affine patches
Let A denote the affine patch of P(H0(Md(V ), L)
∗) given by qv 6= 0. The
origin of the affine space A is identified as the T -fixed point ev. The functions
fθ := qθ/qv, v 6= θ ∈ I(d), provide a set of coordinate functions on A. Monomials
in these fθ form a k-basis for the polynomial ring k[A] of functions on A, where k
denotes the underlying field.
Fix w ≥ v in I(d), so that the point ev belongs to the Schubert varietyX(w),
and let Y (w) be the affine patch of X(w) defined thus:
Y (w) := X(w) ∩ A.
The coordinate ring k[Y (w)] of Y (w) is a quotient of the polynomial ring k[A],
and the proposition that follows identifies a subset of the monomials in fθ which
forms a k-basis for k[Y (w)].
We say that a standard monomial θ1 ≥ . . . ≥ θt in I(d) is v-compatible if for
each k, 1 ≤ k ≤ t, either θk 
 v or v 
 θk. Given w in I(d), we denote by SM
the set of w-dominated v-compatible standard monomials.
Proposition 3.2.1 As θ1 ≥ . . . ≥ θt runs over the set SM
w of w-dominated
v-compatible standard monomials, the elements fθ1 · · · fθt form a basis for the
coordinate ring k[Y (w)] of the affine patch Y (w) = X(w) ∩ A of the Schubert
variety X(w).
Proof: The proof is similar to the proof of Proposition 3.1 of [7]. First consider
a linear dependence relation among the fθ1 · · · fθt . Replacing fθ by qθ and “ho-
mogenizing” by qv yields a linear dependence relation among the w-dominated
standard monomials qθ1 · · · qθs restricted to X(w), and so the original relation
must only have been the trivial one, for by Theorem 3.1.1 the qθ1 · · · qθs are
linearly independent on X(w).
To prove that fθ1 · · · fθt generate k[Y (w)] as a vector space, we make the
following claim: if qµ1 · · · qµr be any monomial in the Pfaffians qθ, and qτ1 · · · qτs
a standard monomial that occurs with non-zero co-efficient in the expression for
(the restriction to X(w) of) qµ1 · · · qµr as a linear combination of w-dominated
standard monomials, then τ1∪· · ·∪τs = µ1∪· · ·∪µr as multisets of {1, . . . , 2d}.
To prove the claim, consider the maximal torus T of SO(V ) as in §1.2. The affine
patch A is T -stable and there is an action of T on k[Y (w)]. The sections qθ are
eigenvectors for T with corresponding characters ǫθ1+· · ·+ǫθd , where ǫk denotes
the character of T given by the projection to the diagonal entry on row k. The
claim now follows since eigenvectors corresponding to different characters are
linearly independent.
Let fµ1 · · · fµr be an arbitrary monomial in the fθ. Fix an integer h such
that h > r(d − 1) and consider the expression for (the restriction to X(w) of)
qµ1 · · · qµr · q
v as a linear combination of w-dominated standard monomials. We
claim that qv occurs in every standard monomial qτ1 · · · qτr+h in this expression
(from which it will follow that the τj are all comparable to v). Suppose that
none of τ1, . . . , τr+h equals v. For each τj there is at least one entry of v that
does not occur in it. The number of occurrences of entries of v in τ1 ∪ · · · ∪ τr+h
is thus at most (r + h)(d − 1). But these entries occur at least hd times in
µ1 ∪ · · · ∪ µr ∪ v ∪ · · · ∪ v (where v is repeated h times), a contradiction to the
claim proved in the previous paragraph. Hence our claim is proved. Dividing
by qr+hv the expression for qµ1 · · · qµr .q
v as a linear combination of w-dominated
standard monomials provides an expression for fµ1 · · · fµr as a linear combina-
tion of fθ1 · · · fθt , as θ1 ≥ . . . ≥ θt varies over SM
3.2.2 Standard monomial theory for tangent cones
The affine patch Md(V )∩A of the orthogonal Grassmannian Md(V ) is an affine
space whose coordinate ring can be taken to be the polynomial ring in variables
of the form X(r,c) with (r, c) ∈ OR, where (as in §2.1)
OR = {(r, c) | 1 ≤ r, c ≤ 2d, r 6∈ v, c ∈ v, r < c∗}
Taking d = 5 and v = (1, 3, 4, 6, 9) for example, a general element of Md(V )∩A
has a basis consisting of column vectors of a matrix of the following form:

1 0 0 0 0
X21 X23 X24 X26 0
0 1 0 0 0
0 0 1 0 0
X51 X53 X54 0 −X26
0 0 0 1 0
X71 X73 0 −X54 −X24
X81 0 −X73 −X53 −X23
0 0 0 0 1
0 −X81 −X71 −X51 −X21

The expression for fθ = qθ/qv in terms of the X(r,c) is a square root of the
determinant of the submatrix of a matrix like the one above obtained by choosing
the rows given by the entries of θ. Thus fθ is a homogeneous polynomial of
degree the v-degree of θ, where the v-degree of θ is defined as one half of the
cardinality of v \ θ.
Since the ideal of the Schubert variety X(w) in the homogeneous coordinate
ring of Md(V ) is generated
2 by the qτ , τ ∈ I(d) such that τ 6≤ w, it follows
that the ideal of Y (w) := X(w) ∩ A in Md(V ) ∩ A is generated by the the fτ ,
τ ∈ I(d) such that τ 6≤ w. We are interested in the tangent cone to X(w)
at ev (or, what is the same, the tangent cone to Y (w) at the origin), and since
k[Y (w)] is graded, its associated graded ring with respect to the maximal ideal
corresponding to the origin is k[Y (w)] itself.
Proposition 3.2.1 says that the graded piece of k[Y (w)] of degree m is gener-
ated as a k-vector space by elements of degreem of the set SMw of w-dominated
v-compatible standard monomials, where the degree of a standard monomial
θ1 ≥ . . . ≥ θt is defined to be the sum of the v-degrees of θ1, . . . , θt. To prove
Theorem 2.3.1 it therefore suffices to prove the following:
2This is a consequence of Theorem 3.1.1. It is easy to see that the qτ such that τ 6≤ w vanish
on X(w). Since all standard monomials form a basis for the homogeneous coordinate ring
of Md(V ) in P(H
0(Md(V ), L)
∗), it follows that w-dominated standard monomials span the
quotient ring by the ideal generated by such qτ . Since such monomials are linearly independent
in the homogeneous coordinate ring of X(w), the desired result follows.
Theorem 3.2.2 The set SMw(m) of standard monomials in I(d) of degree m
that are w-dominated and v-compatible is in bijection with the set Sw(v)(m) of
monomials in OR of degree m that are O-dominated by w.
4 Further reductions
In the last section, we reduced the proof of our main theorem (Theorem 2.3.1)
to that of Theorem 3.2.2. We now reduce the proof of Theorem 3.2.2 to that of
Propositions 4.1.1, 4.1.2 and 4.1.3 below. These propositions will eventually be
proved in §10.
4.1 The main propositions
Fix once and for all an element v of I(d). The bijection stated in Theorem 3.2.2
will be described by means of two maps Oπ and Oφ whose definitions will be
given in §7 and §8 below. We will now state some properties of these maps.
In §4.2 we will see how Theorem 3.2.2 follows once these properties are estab-
lished.
The map Oπ associates to a monomial S in ON a pair (w,S′) consisting
of an element w of I(d) and a “smaller” monomial S′ in ON. This map enjoys
the following good properties:
Proposition 4.1.1 1. w ≥ v.
2. v-degree(w) + degree(S′) = degree(S).
3. w O-dominates S′.
4. w is the least element of I(d) that O-dominates S.
The map Oφ, on the other hand, associates a monomial in ON to a pair (w,T)
consisting of an element w of I(d) with w ≥ v and a monomial T in ON that is
O-dominated by w.
Proposition 4.1.2 The maps Oπ and Oφ are inverses of each other.
For an integer f , 1 ≤ f ≤ 2d, consider the following conditions, the first on
a monomial S in ON, the second on an element w of I(d):
(‡) f is not the row index of any element of S and f⋆ is not the
column index of any element of S.
(‡) f is not an entry of w.
(It is convenient to the use the same notation (‡) for both conditions.)
Proposition 4.1.3 Assume that v satisfies (‡)—all references to (‡) in this
proposition are with respect to a fixed f , 1 ≤ f ≤ 2d.
1. Let w be an element of I(d) with w ≥ v and T a monomial in ON that is
O-dominated by w. If w and T both satisfy (‡), then so does Oφ(w,T).
2. If a monomial S in ON satisfies (‡), then so do the “components” w and
S′ of its image under Oπ.
4.2 From the main propositions to the main theorem
Let us now see how Theorem 3.2.2 follows from the propositions of §4.1. Most
of the following argument runs parallel to its counterparts in the case of the
Grassmannian and symplectic Grassmannian (Propositions 4.1.1 and 4.1.2 have
their counterparts in [7, 4]), but, in the case that d is odd, the part involving
the “mirror image” requires additional work. This is where Proposition 4.1.3
comes in.
Let S, T , and U , denote respectively the sets of monomials in OR, ON, and
OR\ON. Let SMv denote the set of v-compatible standard monomials that are
“anti-dominated” by v: a standard monomial θ1 ≥ . . . ≥ θt is anti-dominated
by v if θt ≥ v (we can also write θt > v since θt 6= v by v-compatibility).
Define the domination map from T to I(d) by sending a monomial in ON to
the least element that O-dominates it. Define the domination map from SMv
to I(d) by sending θ1 ≥ . . . ≥ θt to θ1. Both these maps take, by definition, the
value v on the empty monomial.
Notation 4.2.1 In the following, we use subscripts, superscripts, suffixes, and
combinations thereof to modify the meanings of S, T , U , SM , and SMv.
• superscript: this will be an element w of I(d); when used on T it denotes
O-domination (more precisely, Tw denotes the subset of T consisting of
those elements that are O-dominated by w); when used on SM or SMv
it denotes domination by w.
• subscript: denotes anti-domination (applied only to standard monomials).
• suffix “(m)”: indicates degree (for example, SMwv (m) denotes the set of v-
compatible standard monomials that are anti-dominated by v, dominated
by w, and of degree m).
Repeated application of Oπ gives a map from T to SMv that commutes
with domination (as just defined) and preserves degree. Repeated application
of Oφ gives a map from SMv to T . These two maps being inverses of each
other (Proposition 4.1.2) and so we have a bijection between SMv and T . In
fact, since domination and degree are respected (Proposition 4.1.1), we get a
bijection SMwv (m)
∼= Tw(m).
As explained below, the “mirror image” of the bijection SMv(m) ∼= T (m)
gives a bijection SMv(m) ∼= U(m). Putting these bijections together, we get
the desired result:
SMw(m) =
SMwv (k)× SM
v(m− k)
Tw(k)× U(m− k) = Sw(m).
We now explain how to realize the bijection SMv(m) ∼= U(m) as the “mirror
image” of the bijection SMv(m) ∼= T (m). For an element u of I(d), define
u∗ := (u∗d, . . . , u
1). In the case d is even, the association u 7→ u
∗ is an order
reversing involution, and the argument in [4] for the symplectic Grassmannian
holds here too. In the case d is odd, u∗ is not an element of I(d), and so some
additional work is required.
Recall that a “base element” v of I(d) has been fixed and that our notation
does not explicitly indicate this dependence upon v: for example, OR is depen-
dent upon v. For a brief while now (until the end of this section) we need to
simultaneously handle several base elements of I(d). We will use the following
convention: when the base element of I(d) is not v, we will explicitly indicate
it by means of a suffix. For instance, SM(v∗) denotes the set of v∗-compatible
standard monomials in I(d).
Let us first do the case when d is even. We get a bijection SMv ∼= SMv∗(v
by associating to θ1 ≥ . . . ≥ θt the element θ
t ≥ . . . ≥ θ
1 . The sum of the
v-degrees of θ1, . . . , θt equals the sum of the v
∗-degrees of θ∗t , . . . , θ
1 , so that we
get a bijection SMv(m) ∼= SMv∗(v
∗)(m).
For an element (r, c) of ON(v∗), consider its flip (c, r). Since v belongs
to I(d), the complement of v∗ in {1, . . . , 2d} is v, and it follows that (c, r)
belongs to OR \ ON. This induces a degree preserving bijection T (v∗) ∼= U .
Putting this together with the bijection of the previous paragraph and the one
deduced earlier in this section (using Oπ and Oφ), we get what we want:
SMv(m) ∼= SMv∗(v
∗)(m) ∼= T (v
∗)(m) ∼= U(m).
Now suppose that d is odd. Then the map x 7→ x∗ does not map I(d) to
I(d) but to I(d)∗ (defined as the set consisting of those elements u of I(d, 2d)
such that, for each k, 1 ≤ k ≤ 2d, exactly one of k, k∗ belongs to u, and the
number of entries of u greater than d is odd). We define a map u 7→ ũ from
I(d)∗ to I(d + 1) as follows: ũ := {ũ1, . . . , ũd, d + 2} (the elements are not in
increasing order except in the trivial case u = (1, . . . , d)), where, for an integer
e, 1 ≤ e ≤ 2d, we set
ẽ :=
e if 1 ≤ e ≤ d
e+ 2 if d+ 1 ≤ e ≤ 2d
This map u 7→ ũ is an order preserving injection.
Consider the composition x 7→ x∗ 7→ x̃∗ from I(d) to I(d+1). This is an order
reversing injection. The induced map on standard monomials is an injection
from SMv to SMfv∗(ṽ
∗). It is readily seen that the image under this map
is the subset SMfv∗(ṽ
∗)(‡) consisting of those standard monomials all of whose
elements satisfy (‡) with f = d+1. We have already established (using the maps
Oπ and Oφ) a bijection SMfv∗(ṽ
∗) ∼= T (ṽ∗). It follows from Proposition 4.1.3
that under this bijection the subset SMfv∗(ṽ
∗)(‡) maps to T (ṽ∗)(‡) (defined as
the set of those monomials in ON(ṽ∗) satisfying (‡) with f = d+ 1).
Now T (ṽ∗)(‡) is in degree preserving bijection with U : every element of
degree 1 of T (ṽ∗)(‡) is uniquely of the form (c̃, r̃) for (r, c) in OR\ON, and the
desired bijection is induced from this. Putting all of these together, we finally
SMv ∼= SMfv∗(ṽ
∗)(‡) ∼= T (ṽ∗)(‡) ∼= U.
Thus, in order to prove our main theorem (Theorem 2.3.1), it suffices to
describe the maps Oπ and Oφ and to prove Propositions 4.1.1–4.1.3.
Part III
The proof
The main combinatorial results formulated in §4.1 are proved. An attempt is
made to maintain parallelism with the proofs in [7].
5 Terminology and notation
5.1 Distinguished subsets
5.1.1 Distinguished subsets of N
Following [7, §4], we define a multiset S of N to be distinguished , if, first of all,
it is a subset in the usual sense (in other words, it is “multiplicity free”), and
if, for any two distinct elements (R,C) and (r, c) of S, the following conditions
are satisfied:
A. R 6= r and C 6= c.
B. If R > r, then either r < C or C < c.
In terms of pictures, condition A says that (r, c) cannot lie exactly due North
or East of (R,C) (or the other way around); so we can assume, interchanging
the two points if necessary, that (r, c) lies strictly to the Northeast or Northwest
of (R,C); condition B now says that, if (r, c) lies to the Northwest of (R,C),
then the point that is simultaneously due North of (R,C) and due East of (r, c)
(namely (r, C)) does not belong to N.
5.1.2 Attaching elements of I(d, 2d) to distinguished subsets of N
To a distinguished subset S of N there is naturally associated an element w of
I(d, 2d) as follows: start with v, remove all members of v which appear as column
indices of elements of S, and add row indices of all elements of S. As observed
in [7, Proposition 4.3], this association gives a bijection between distinguished
subsets of N and elements w ≥ v of I(d, 2d). The unique distinguished subset
of N corresponding to an element w ≥ v of I(d, 2d) is denoted Sw.
5.2 The involution #
5.2.1 The involution # on I(d, 2d)
There are two natural order reversing involutions on I(d, 2d). First there is
w 7→ w∗ induced by the natural order reversing involution j 7→ j∗ on {1, . . . , 2d}:
here w∗ has the obvious meaning, namely, it consists of all j∗ such that j
belongs to w. Then there is the map taking w to its complement {1, . . . , 2d}\w.
These two involutions commute. Composing the two we get an order preserving
involution on I(d, 2d) which we denote by w 7→ w#. The elements of the
subset I(d) are fixed points under this involution (there are points not in I(d)
that are also fixed).
5.2.2 The involution # on N and R
For α = (r, c) in N, or more generally in R, define α# = (c∗, r∗). The involution
α 7→ α# is just the reflection with respect to the diagonal d. For a subset or
even multiset S of N (or R), the symbol S# has the obvious meaning. We call
S symmetric if S = S#.
Proposition 5.2.1 An element w ≥ v of I(d, 2d) belongs to I(d) if and only
if the distinguished subset Sw of N corresponding to it as described in §5.1.2 is
symmetric and has evenly many diagonal elements.
Proof: That the symmetry of Sw is equivalent to the condition that w = w
is proved in [4, Proposition 5.7]. Now suppose that Sw is symmetric. We claim
that for an element (r, c) of Sw that is not on the diagonal, either both r and c
are bigger than d or both are less than d + 1. It is enough to prove the claim,
for w is obtained from v by removing the column indices and adding the row
indices of elements of Sw, and it would follow that the number of entries in w
that are bigger than d equals the number of such entries in v plus the number
of diagonal elements in Sw.
We now prove the claim. Since Sw is symmetric, it follows that (c
∗, r∗) also
belongs to Sw. Since Sw is distinguished, it follows that in case r < c
∗ (that
is, if (r, c) lies above the diagonal), we have r < r∗, and so c < r < r∗; and in
case r > c∗, we have c∗ < c, and so c∗ < c < r. Thus the claim is proved. �
5.3 The subset SC attached to a v-chain C
5.3.1 Vertical and horizontal projections of an element of ON
For α = (r, c) in ON (or more generally in OR), the elements pv(α) := (c
∗, c)
and ph(α) := (r, r
∗) of the diagonal d are called respectively the vertical and
horizontal projections of α. In terms of pictures, the vertical projection is the
element of the diagonal due South of α; the horizontal projection is the element
of the diagonal due East of α. The vertical line joining α to its vertical projection
pv(α) and the horizontal line joining α to its horizontal projection ph(α) are
called the legs of α.
5.3.2 The “connection” relation on elements of a v-chain
Let C : α1 = (r1, c1) > α2 = (r2, c2) > · · · be a v-chain in ON. Two consecutive
elements αj and αj+1 of C are said to be connected if the following conditions
are both satisfied:
• their legs are “intertwined”; equivalently and more precisely, this means
that r∗j ≥ cj+1, or, what amounts to the same, rj ≤ c
• the point (rj+1, r
j ) belongs to N; this just means that rj+1 > r
Consider the coarsest equivalence relation on the elements of C generated by
the above relation. The equivalence classes of C with respect to this equivalence
relation are called the connected components of the v-chain C.
This definition has its quirks:
The v-chain C : α > β > γ in the pic-
ture has {α, β} and {γ} as its connected
components; but the “sub” v-chain α > γ
of C is connected (as a v-chain in its own
right).
diagonal
boundary
5.3.3 The definition of SC
We will define SC as a multiset of N. It is easy to see and in any case stated
explicitly as part of Corollary 5.3.5 that it is multiplicity free and so is actually
a subset of N.
First suppose that C : α1 = (r1, c1) > · · · > αℓ = (rℓ, cℓ) is a connected
v-chain in ON. Observe that, if there is at all an integer j, 1 ≤ j ≤ ℓ, such that
the horizontal projection ph(αj) does not belong to N, then j = ℓ. Define
SC :=
{pv(α1), . . . , pv(αℓ)} if ℓ is even
{pv(α1), . . . , pv(αℓ)} ∪ {ph(αℓ)} if ℓ is odd and ph(αℓ) ∈ N
{pv(α1), . . . , pv(αℓ−1)} ∪ {αℓ, α
} if ℓ is odd and ph(αℓ) 6∈ N
For a v-chain C that is not necessarily connected, let C = C1 ∪ C2 ∪ · · · be
the partition of C into its connected components, and set
SC := SC1 ∪SC2 ∪ · · ·
5.3.4 The type of an element α of a v-chain C, and the set SC,α
We introduce some terminology and notation. Their usefulness may not be
immediately apparent.
Suppose that C : α1 > · · · > αℓ is a connected v-chain. We define the type
in C of an element αj , 1 ≤ j ≤ ℓ, of C to be V, H, or S, accordingly as:
V: j 6= ℓ, or j = ℓ and ℓ is even.
H: j = ℓ, ℓ is odd, and ph(αℓ) ∈ N.
S: j = ℓ, ℓ is odd, and ph(αℓ) 6∈ N.
The type of an element in a v-chain that is not necessarily connected is defined
to be its type in its connected component.
The set SC,α of elements of N generated by an element α of C is defined to
SC,α :=
{pv(α)} if α is of type V in C;
{pv(α), ph(α)} if α is of type H in C;
{α, α#} if α is of type S in C;
Observe that, for a v-chain C, the monomial SC defined in §5.3.3 is the union,
over all elements α of C, of SC,α.
For an element α of a v-chain C, we define qC,α to be pv(α) if α is of type V
or H and to be α if it is of type S.
If the horizontal projection of an element in a v-chain does not belong to N,
then clearly the same is true for every succeeding element. The first such element
of a v-chain is called the critical element.
Proposition 5.3.1 1. The cardinality is odd of a connected component that
has an element of type H or S. Conversely, if the cardinality of a compo-
nent is odd, then it has an element of type H or S.
2. An element of type H or S can only be the last element in its connected
component.
3. The critical element has type either V or S. No element before it can be of
type S and every element after it is of type S. In particular, any element
that succeeds an element of type S is of type S.
Proof: Clear from definitions. �
Proposition 5.3.2 Let α > γ be elements of a v-chain C (we are not assuming
that they are consecutive).
1. If α > γ is connected as a v-chain in its own right, then α is connected to
its next member in C; that is, α cannot be the last element in its connected
component in C.
2. If α > γ is not connected as a v-chain in its own right and the legs of α and
γ intertwine, then the connected component of γ in C is the singleton {γ},
and γ has type S in C.
Proof: Clear from definitions. �
Proposition 5.3.3 Let E : α > . . . > ζ be a v-chain, D and D′ two v-chains
with tail α, and C, C′ the concatenations of D, D′ respectively with E. Then
1. The last element in the connected component containing α is the same
in C and C′ (and this is the same as in E).
Let λ denote this element.
2. The only element among α, . . . , ζ that possibly has different types in C
and C′ is λ.
Proof: (1): Whether or not two successive elements in a v-chain are connected
is independent of other elements in the v-chain.
(2): The type of an element in a v-chain is V unless it is the last element
in its connected component. And the type of the last element in a component
depends on the cardinality of the component. The components of E not contain-
ing α are still components in C and C′. In contrast, the component containing α
could possibly be larger in C (respectively C′) and hence its cardinality could
be different. �
For an element α = (r, c) of N, we define α(up) to be α itself if α is either on
or above the diagonal d (more precisely, if r ≤ c∗), and to be its “reflection” in
the diagonal (more precisely, (c∗, r∗)) if α is below the diagonal (more precisely,
if r > c∗). For a monomial S of N, S(up) is defined to be the intersection
of S (as a multiset) with the subset ON ∪ d of N. The notations α(down) and
S(down) have similar meanings.
Caution: It is not true that S(up) = {α(up)|α ∈ S} (in the obvious sense one
would make of the right hand side). In particular, for a singleton monomial {α},
it is not always true that {α}(up) = {α(up)}.
Proposition 5.3.4 Let α and β be elements of a v-chain C. Let us use α′ and
β′ respectively to denote elements of SC,α(up) and SC,β(up).
1. If α > β (these elements are not necessarily consecutive in C), then, given
β′, there exists α′ such that α′ > β′. In fact, this is true for every choice
of α′ except when
(*) α is of type H, and ph(α) 6> β
′ for some β′ ∈ SC,β.
In particular, qC,α > β
′ and qC,α > qC,β.
2. Conversely, suppose that α′ > β′ for some choice of α′ and β′. Then
α ≥ β; if equality occurs, then α is of type H, α′ = pv(α) and β
′ = ph(α).
In particular, if α′ > qC,β (or more specially qC,α > qC,β), then α > β.
3. If (*) holds for α > β in C, then
(a) the critical element of C is the one just after α; in particular, α is
uniquely determined.
(b) all elements of C succeeding α are of type S; in particular, β is of
type S and β′ = β.
(c) (*) holds for γ in place of β for every γ in C that succeeds α.
Proof: (1) If α is of type V or H, we need only take α′ = pv(α), for pv(α) >
pv(β), pv(α) > ph(β), and pv(α) > β. Now suppose that α is of type S. Then β
too is of type S (Proposition 5.3.1 (3)), so β′ can only be β, and the first part
of (1) is proved.
It follows from the above that if α′ = pv(α) or if α has type S, then α
′ > β′
independent of the choice of α′. So if α′ 6> β′, then (*) holds and α′ = ph(α).
(3) Let λ be the immediate successor of α in C. Then α is not connected
to λ (Proposition 5.3.1 (2)). Since ph(α) 6> β
′, it follows that α and β have
intertwining legs. Therefore so do α and λ. By Proposition 5.3.2 (2), λ has
type S in C.
Since α has type H and λ type S, it follows immediately from the definition of
the critical element that λ is the critical element. This proves (a). Assertion (b)
now follows from Proposition 5.3.1 (3). For (c), write ph(α) = (a, a
∗), λ =
(R,C), and γ = (r, c). Then R < a∗, for α and λ have intertwining legs but
are not connected. So c < r ≤ R < a∗. This means ph(α) 6> γ. And γ being of
type S (by (b)), we can take γ′ = γ.
(2) Suppose that α 6≥ β. Then β > α. By the second part of (1) above, β
is of type H and β′ = ph(β); by item (b) of (3), α is of type S, so α
′ = α. This
leads to the contradiction β > α > ph(β). �
Corollary 5.3.5 The multiset SC attached to a v-chain C is a distinguished
subset of N in the sense of 5.1.1.
Proof: If α in C is of type V or S, then SC,α is a singleton; if it is of type H,
then SC,α = {pv(α), ph(β)}. So there can be no violation of conditions A and B
of §5.1.1 by elements of SC,α.
Suppose α > β. By Proposition 5.3.4 (1), we have α′ > β′ for any choice
of α′ ∈ SC,α and β
′ ∈ SC,β except when the condition (*) holds. By (3)
of the same proposition, if (*) holds, then β′ = β, and writing β = (r, c),
ph(α) = (a, a
∗), we have r < a (since α > β) and c < r < a∗ (see proof of
item 3(c) of the proposition). Thus there can be no violation of conditions A
and B of §5.1.1. �
Corollary 5.3.6 Let S be a v-chain in ON and w an element of I(d). If w
O-dominates S, then w dominates in the sense of [7] the monomial S ∪ S#
of N.
Proof: By [4, Proposition 5.15], it is enough to show that w dominates S. Let
C : α1 > . . . > αt be a v-chain in S. Writing αj = (rj , cj) and qC,αj = (Rj , Cj)
we have rj ≤ Rj and Cj ≤ cj . By Proposition 5.3.4 (1), we have qC,α1 > . . . >
qC,αt . Since w O-dominates S, it in particular dominates qC,α1 > . . . > qC,αt
and so also C. �
6 O-depth
The concept of O-depth defined in §6.1 below plays a key role in this paper. As
the name suggests, it is the orthogonal analogue of the concept of depth of [7].
In §6.2 below, it is observed that the O-depth is no smaller than depth in the
sense of [7]. In §6.3, some observations about the relation between O-depths
and types of elements in v-chains are recorded.
6.1 Definition of O-depth
The O-depth of an element α in a v-chain C in ON is the depth in SC in the
sense of [7] of qC,α: in other words, it is the depth in SC of pv(α) in case α is of
type V or H, and of α (equivalently of α#) in case α is of type S. It is denoted
O-depthC(α). The O-depth of an element α in a monomial S of ON is the
maximum, over all v-chains C in S containing α, of the O-depth of α in C. It
is denoted O-depth
(α). Finally, the O-depth of a monomial S in ON is the
maximum of the O-depths in S of all the elements of S.
There is a conflict in the above definitions: Is the O-depth of an element
of a v-chain C the same as its depth as an element of the monomial C? In
other words, could the O-depth of an element in a v-chain be exceeded by its
O-depth in a sub-chain? The conflict is resolved by the first item of the following
proposition.
Proposition 6.1.1 1. For v-chains C ⊆ D, the O-depth in C of an element
of C is no more than its O-depth in D.
2. If a v-chain C is an initial segment of a v-chain D, then the O-depths
in C and D of an element of C are the same.
Proof: (1): By an induction on the difference in the cardinalities of D and C,
we may assume that D has one more element than C. Call this extra element δ.
Suppose that δ lies between successive elements α and β of C (the modifications
needed to cover the extreme cases when it goes at the beginning or the end are
being left to the reader).
The only elements of C that could possibly undergo changes of type on
addition of δ are α and the last element in the connected component of β,
which let us call β′. If there are no type changes, then SC ⊆ SD and the
assertion is immediate. The only type change that α can undergo is from H
to V. The type changes that β′ can undergo are: H to V; V to H; S to V; V to
S. An easy enumeration of cases shows that only one of α and β′ can undergo
a type change.
We need not worry about changes from V to H for in this case SC ⊆ SD.
First let us suppose that α undergoes a change of type (from H to V). Then
δ is connected to α. It follows from Proposition 5.3.1 (1) that δ has type V in D:
the connected component of α in C has odd number of elements, so if δ happens
to be the last element in its connected component in D, the number of elements
in that component will be even. Replacing an occurrence of ph(α) in a v-chain
of SC by pv(δ) would result in a v-chain in SD (by Proposition 5.3.4 (1)), and
this case is settled.
Now suppose that β′ undergoes a type change. Then δ is connected to β and
δ is of type V in D (Proposition 5.3.1 (2)). Replacing by pv(δ) any occurrence
in a v-chain in SC of pv(β
′), ph(β
′), β′ accordingly as the type of β′ in C
is V, H, or S, (not necessarily in the same place but at an appropriate place)
would result in a v-chain in SD (by Proposition 5.3.4 (1)), and we see that the
O-depth cannot decrease.
(2): It follows from Proposition 5.3.4 (2) that, for an element α of C, con-
tributions to SD from elements beyond α (in particular from those not in C)
do not affect the depth in SD of qD,α. Looking for the possibility of differences
in types in C and D of elements of C, we see that the only element of C that
has possibly a different type in D is its last element. And this too can change
type only from H to V.
The above two observations imply that the calculations of O-depths in C
and D of an element α of C are no different: we would be considering the depth
in SC and SD respectively of the same element (either pv(α) or α), and the
differences in SD and SC have no effect on this consideration. �
Corollary 6.1.2 If C ⊆ D are v-chains in ON, then wC ≤ wD (although it is
not always true that SC ⊆ SD).
Proof: By [7, Lemma 5.5], it is enough to show that every v-chain in SC is
dominated by wD. Let β1 = (r1, c1) > · · · > βt = (rt, ct) be an arbitrary v-chain
in SC . To show that it is dominated by wD, it is enough, by [7, Lemma 4.5], to
show the existence of a v-chain (R1, C1) > · · · > (Rt, Ct) in SD with rj ≤ Rj
and Cj ≤ cj for 1 ≤ j ≤ t. Such a v-chain exists by the proof of (1) of Proposi-
tion 6.1.1. �
Corollary 6.1.3 1. Let S be a monomial in ON and α ∈ S. Then there ex-
ists a v-chain C in S with tail α such that O-depth
(α) = O-depthC(α).
2. For elements α > γ in a v-chain C (these need not be consecutive), we
have O-depthC(α) < O-depthC(γ).
3. For elements α > γ of a monomial S in ON, we have O-depth
(α) <
O-depth
4. No two elements of the same O-depth in a monomial in ON are compa-
rable.
Proof: (1) This follows from (2) of the Proposition above and the definition
of O-depth.
(2) This follows from Proposition 5.3.4 (1) and the definition of O-depth.
(3) By (1), there exists a v-chain C with tail α such that O-depth
(α) =
O-depthC(α). Concatenate C with α > γ and letD denote the resulting v-chain.
By (2) of the Proposition above, O-depthC(α) = O-depthD(α). By (2) above,
O-depthD(α) < O-depthD(γ). And finally, O-depthD(γ) ≤ O-depthS(γ) by
the definition of O-depth
(4) Immediate from (3). �
Corollary 6.1.4 Let β > γ be elements of a v-chain C of elements of ON. Let
E be a v-chain in SC with tail qC,γ and length O-depthC(γ). Then qC,β occurs
in E.
Proof: It is enough to show that for α′ 6= qC,β in E, either α
′ > qC,β or
qC,β > α
′. Let α be in C such that qC,β 6= α
′ ∈ SC,α. If β ≥ α, then qC,β > α
by Proposition 5.3.4 (1). If α > β and α′ 6> qC,β , then, by (1) and (3) of the
same proposition, α′ 6> qC,γ , a contradiction. �
6.2 O-depth and depth
Lemma 6.2.1 The O-depth of an element α in a monomial S of ON is no
less than its depth (in the sense of [7]) in S ∪S#.
Proof: Let C : α1 > . . . > αt be a v-chain in S∪S
# with tail αt = α, where t
is the depth of α in S ∪S#. We then have α1(up) > . . . > αt(up), so we may
assume C to be in S. By Proposition 5.3.4 (1), qC,α1 > . . . > qC,αt in SC . So
depth
S∪S#(α) = t ≤ depthSC (qC,αt) ≤ O-depthS(α). �
6.3 O-depth and type
We begin by defining some useful terminology. Let (r, c) and (R,C) be two
elements of R. To say that (R,C) dominates (r, c) means that r ≤ R and C ≤ c
(in terms of pictures, (r, c) lies (not necessarily strictly) to the Northeast of
(R,C)). To say that they are comparable means that either (R,C) > (r, c) or
(r, c) > (R,C). While this is admittedly strange, there will arise no occasion
for confusion.
For an integer i, we let i(odd) be the largest odd integer not bigger than i
and i(even) the smallest even integer not smaller than i.
Lemma 6.3.1 1. For consecutive elements α > β of a v-chain C,
O-depthC(β) =
O-depthC(α) + 2 if and only if α is of
type H and ph(α) > β
O-depthC(α) + 1 otherwise
2. For an element of a v-chain C such that either its horizontal projection
belongs to N or it is connected to its predecessor, the parity of its O-depth
in C is the same as that of its ordinality in its connected component in C.
3. The O-depth in a v-chain of an element of type H is odd.
4. If in a v-chain an element of type V is the last in its connected component,
then its O-depth is even.
5. If in a v-chain C there is an element of O-depth d, then
(a) for every odd integer d′ not exceeding d, there is in C an element of
O-depth d′.
(b) if, for an even integer d′ not exceeding d, there is no element in C of
O-depth d′, then the element α in C of O-depth d′ − 1 is of type H,
and ph(α) > β, where β denotes the immediate successor of α in C.
6. Let C be a v-chain and α an element of type H in C. Then the depth in
SC of ph(α) equals O-depthC(α) + 1. In particular, this depth is even.
Proof: (1): From items 1 and 3(a) of Proposition 5.3.4, it follows that, for
γ in C with γ > α, if γ′ 6> qC,α for some γ
′ in SC,γ , then γ
′ 6> qC,β. Thus
O-depthC(β) exceeds O-depthC(α) by the number of elements in SC,α that
dominate qC,β . This number is 1 if α is of type V, or of type S, or of type H
and ph(α) 6> β; it is 2 if α is of type H and ph(α) > β (note that ph(α) > β if
and only if ph(α) > qC,β).
(2): Let λ be such an element. Everything preceding λ in C is of type H
or V (Proposition 5.3.1 (3)). Let λ belong to the kth connected component,
and n1, . . . , nk be respectively the cardinalities of the first, . . . , k
th connected
components. By (1) above and item 3(b) of Proposition 5.3.4, O-depthC(λ)
is n1(even) + · · · + nk−1(even) plus the ordinality of λ in the k
th connected
component.
(3) and (4): These are special cases of 2.
(5): This follows easily from (1) and (3).
(6): It follows from Proposition 5.3.4 (2) that there is no element γ in SC
that lies between pv(α) and ph(α) (meaning pv(α) > γ > ph(α)), so the asser-
tion holds. �
Corollary 6.3.2 For a v-chain C in ON, if the O-depths of elements in C are
bounded by k, then the depths of elements in SC are bounded by k(even).
Proof: The depth of qC,α in SC for any α in C is at most k by hypothesis.
An element of SC that is not qC,α for any α in C can only be of the form
ph(α) for some α. By Proposition 5.3.4, depthSCpv(α) = depthSCph(α) − 1,
which implies depth
ph(α) ≤ k + 1. If, moreover, k is even, then by (3) of
Lemma 6.3.1 depth
ph(α) = depthSCpv(α) + 1 ≤ (k − 1) + 1 = k. �
Proposition 6.3.3 Given a monomial S in ON and an element α in it, there
exists a v-chain C in S with tail α such that O-depthC(β) = O-depthS(β) for
every β in C.
Proof: Proceed by induction on d := O-depth
(α). Choose a v-chain D in S
with tail α such that O-depthD(α) = O-depthS(α) (such a v-chain exists by
Corollary 6.1.3 (1)). Let α′ be the element in D just before α. It follows from
item (3) of Corollary 6.1.3 and item (1) of Lemma 6.3.1 that O-depth
(α′) (as
also O-depthD(α
′)) is either d − 1 or d − 2. By induction, there exists a v-
chain C′ with tail α′ that has the desired property. Let C be the concatenation
of C′ with α′ > α.
We claim that C has the desired property. The only thing to be proved is
that O-depthC(α) = d. By item (1) of Lemma 6.3.1, we have O-depthC(α) ≥
O-depthC′(α
′) + 1. In particular, the claim is proved in case O-depthC′(α
is d − 1, so let us assume that O-depthC′(α
′) is d− 2. It now follows from the
same item that α′ has type H in D and ph(α
′) > α; it further follows that it is
enough to show that α′ has type H in C.
Since α′ has type H in D, it follows (from item (2) of Proposition 5.3.1) that
α′ > α is not connected and (from item (3) of Lemma 6.3.1) that d− 2 is odd.
Now, by item (4) of Lemma 6.3.1, the type in C′ of α′ cannot be V, so it is H,
and the claim is proved. �
Corollary 6.3.4 Let S be a monomial in ON, β an element of S, and i an
integer such that i < O-depth
(β). Then
(a) If i is odd, there exists an element α in S of O-depth i such that α > β.
(b) If i is even and there is no element α in S of O-depth i such that α > β,
then there is element α in S of O-depth i− 1 such that ph(α) > β.
Proof: Choose a v-chain C in S having tail β and the good property of Propo-
sition 6.3.3. Apply Lemma 6.3.1 (5). �
Corollary 6.3.5 Let C be a v-chain in ON with tail α such that O-depthC(α)
is odd. Let A be a v-chain in ON with head α, and D the concatenation of C
with A. Let C′ denote the v-chain C \ {α}. Then
1. The type of an element of A is the same in both A and D. In particular,
SA ⊆ SD and qA,β = qD,β for β in A.
2. The type of an element of C′ is the same in both C′ and D. In particular,
SC′ ⊆ SD.
3. SD = SC′ ∪ SA (disjoint union); letting j0 := O-depthC(α) we have
j0 = SA and (SD)1 ∪ · · · ∪ (SD)j0−1 = SC′ . (For a monomial S,
the subset of elements of depth at least i is denoted Si, and the subset of
elements of depth exactly i is denoted Si.)
Proof: (1) Generally (meaning without the assumption that O-depthC(α) is
odd), the only element of A that could possibly have a different type in D is
the last one in the first connected component of A; whether or not it changes
type depends exactly upon whether or not the parity of the cardinality of its
connected component in D is different from that in A. Under our hypothesis,
this parity does not change, for, by (4) of Lemma 6.3.1, the type of α in C is H
or S, and so the cardinality of the connected component of α in C is odd.
(2) Generally (meaning without the assumption that O-depthC(α) is odd),
the only element of C′ that could possibly have a different type in D is the last
one of C′; it changes type if and only if it is connected to α and the cardinality
of its connected component in C′ is odd. Under our hypothesis, this cardinality
is even, for the same reason as in (1).
(3) That SD = SC′ ∪ SA (disjoint union) is an immediate consequence
of (1) and (2). By Lemma 6.3.1 (1), qA,α = qD,α dominates every element of
SA, so SA ⊆ (SD)
j0 (depth
qD,α = O-depthD(α) = O-depthC(α) = j0).
It is enough to prove the following claim: every element of SC′ has depth less
than j0 in SD. Let γ
′ be an element of SC′ . If γ
′ > qD,α then the claim is
clear. If not, then, by Proposition 5.3.4 (1), γ′ = ph(γ). By Lemma 6.3.1 (3),
O-depthD(γ) is odd. Since the claim is already true for qD,γ = pv(γ), we
have O-depthD(γ) = depthSDpv(γ) ≤ j0 − 2. By (6) of the same lemma,
depth
γ′ = O-depthD(γ)+1, so depthDγ
′ ≤ j0−1, and the claim is proved.�
Proposition 6.3.6 Let S be a monomial in ON and j an odd integer. For β
in Sj,j+1(:= {α ∈ S |O-depth
(α) ≥ j}), we have
O-depth
Sj,j+1
(β) = O-depth
(β)− j + 1
Proof: Proceed by induction on j. For j = 1, the assertion reduces to a
tautology. Suppose that the assertion has been proved upto j. By the induction
hypothesis, we have Sj+2,j+3 = (Sj,j+1)3,4, and we are reduced to proving the
assertion for j = 3.
Let A be a v-chain in S3,4 with tail β and O-depthA(β) = O-depthS3,4(β).
Let α be the head of A. We may assume that O-depth
(α) = 3 for, if
O-depth
(α) > 3, we can find, by Lemma 6.3.1 (5), α′ of O-depth 3 in S
with α′ > α, and extending A by α′ will not decrease the O-depth in A of β
(Proposition 6.1.1 (1)). Let E be a v-chain in SA with tail qA,β and length
O-depthA(β). The head of E is then qA,α (see Proposition 5.3.4 (1)).
Choose C in S with tail α such that O-depthC(α) = 3. Let D be the
concatenation of C with A. By Corollary 6.3.5, E is contained in SD, qD,α =
qA,α, and qD,β = qA,β. By Proposition 6.1.1 (2), the O-depth of α is the same
in D as in C. Choose a v-chain F in SD with tail qD,α = qA,α. Concatenating F
with E we get a v-chain inSD with tail qD,β = qA,β of lengthO-depthS3,4(β)+2.
This proves that O-depth
(β) ≥ O-depth
(β) + 2.
To prove the reverse inequality, we need only turn the above proof on
its head. Let D be a v-chain in S with tail β such that O-depth
(β) =
O-depthD(β). Let G be a v-chain in SD with tail qD,β and length O-depthS(β).
There exists an element α in D of O-depth 3 in D (by Lemma 6.3.1 (5)). Let
C be the part of D upto and including α, and A the part α > . . . > β. By
Proposition 6.1.1 (2), O-depthC(α) = 3 and, as above, Corollary 6.3.5 applies.
By Corollary 6.1.4, qA,α = qD,α occurs in G. The part F of G upto and in-
cluding qA,α is of length at most 3, and the part E : qD,α > . . . > qD,β belongs
also to SA (Proposition 5.3.4 (2)). Thus the length of G is at most 2 more than
the the length of E which is at most O-depth
(β). �
Corollary 6.3.7 For odd integers i, j, we have (Si,i+1)j,j+1 = Si+j−1,i+j . �
Corollary 6.3.8 Let E : α > . . . > ζ be a v-chain, D and D′ two v-chains with
tail α, and C, C′ the concatenations of D, D′ respectively with E. Then
1. O-depthC(ζ)−O-depthC(α) ≤ O-depthC′(ζ) −O-depthC′(α) + 1;
2. equality holds if and only if the type of λ is H in C and V in C′, and
ph(λ) > µ, where λ is the last element in the connected component con-
taining α of E and µ is the immediate successor in E of λ.
Proof: These assertions follow from combining (2) of Proposition 5.3.3 with
(1) of Lemma 6.3.1. �
Corollary 6.3.9 Let ζ be an element of a monomial S in ON. Let C be a
v-chain in S with tail ζ such that O-depthC(ζ) = O-depthS(ζ). Then
1. O-depthC(α) ≥ O-depthS(α) − 1 for any α in C.
2. If O-depthC(α) = O-depthS(α) − 1 for some α in C, then
(a) letting λ be the last element in the connected component containing α
and µ the element next to λ, the type of λ in C is H and ph(λ) > µ.
(b) O-depthC(γ) = O-depthS(γ) − 1 for all γ in C between α and λ
(both inclusive).
Proof: (1) Let α be in C. Let E denote the part of C beyond (and including) α.
Let D′ be a v-chain in S with tail α such that O-depthD′(α) = O-depthS(α).
Let C′ be the concatenation of D′ and E. Applying Proposition 6.3.8 (1), we
O-depthC(α) ≥ O-depthC(ζ) −O-depthC′(ζ) +O-depthC′(α)− 1.
But O-depthC(ζ)−O-depthC′(ζ) = O-depthS(ζ)−O-depthC′(ζ) ≥ 0, and, by
the choice of D′ and Proposition 6.1.1 (2), O-depthC′(α) = O-depthD′(α) =
O-depth
(2) Assertions (a) and (b) follow respectively from the “only if ” and “if”
parts of item (2) of Proposition 6.3.8. �
7 The map Oπ
The purpose of this section is to describe the map Oπ. The description is given
in §7.1. It relies on certain claims which are proved in §§7.3, 7.4. Those proofs
in turn refer to results from §9, but there is no circularity—to postpone the
definition of Oπ until all the results needed for it have been proved would hurt
rather than help readability. The observations in §7.5 are required only in §10.
The symbol j will be reserved for an odd positive integer throughout this
section.
7.1 Description of Oπ
The map Oπ takes as input a monomial S in ON and produces as output
a pair (w,S′), where w is an element of I(d) such that w ≥ v and S′ is a
“smaller” monomial, possibly empty, in ON. If the input S is empty, no output
is produced (by definition). So now suppose that S is non-empty.
We first partition S into subsets according to the O-depths of its elements.
Let S
be the sub-monomial of S consisting of those elements of S that have
O-depth k—the superscript “pr” is short for “preliminary”. It follows from
Corollary 6.1.3 (4) that there are no comparable elements in S
and so we can
arrange the elements of S
in ascending order of both row and column indices.
Let σk be the last element of S
in this arrangement.
Let now j be an odd integer. We set
j,j+1 := S
We say that S is truly orthogonal at j if ph(σj) belongs to N (that is, if r > r
where σj = (r, c)),
Let Sj,j+1 denote the monomial in N defined by Sj,j+1 :=
j,j+1 \ {σj}
j,j+1 \ {σj}
∪ {pv(σj), ph(σj)}
if S is truly
orthogonal at j
j,j+1 ∪
j,j+1
otherwise
Here S
j,j+1 \ {σj} and other terms on the right are to be understood as mul-
tisets. As proved in Corollary 7.3.4 (1) below, Sj,j+1 has depth at most 2.
Let Sj (respectively Sj+1 be the subset (as a multiset) of elements of depth 1
(respectively 2) of Sj,j+1.
Now, for every integer k, we apply the map of π of [7, §4] to Sk to obtain a
pair (w(k),S′k), where w(k) is an element of I(d, 2d) andS
k is a monomial in N.
Let Sw(k) be the distinguished monomial in N associated to w(k)—see §5.1.2.
Proposition 7.1.1 1. Sw(k) and S
k are symmetric. And therefore so are
∪kSw(k) and ∪kS
2. ∪kSw(k) is a distinguished subset of N (in particular, the Sw(k) are dis-
joint).
3. For j an odd integer, either
• both Sw(j) and Sw(j+1) meet the diagonal, or
• neither of them meets the diagonal,
precisely as whether or not S is truly orthogonal at j. And therefore
∪kSw(k) has evenly many diagonal elements.
4. No S′k intersects the diagonal. And therefore neither does ∪kS
The proposition will be proved below in §7.4.
Finally we are ready to define the image (w,S′) of S under Oπ. We let w
be the element of I(d, 2d) associated to the distinguished subset ∪kSw(k) of N;
since ∪kSw(k) is symmetric and has evenly many diagonal elements, it follows
from Proposition 5.2.1 that w is in fact an element of I(d). And we take S′ :=
k ∩ON.
Remark 7.1.2 Setting
π(Sj,j+1) := (wj,j+1,S
j,j+1), S
′ := ∪j oddS
j,j+1 ∩ON,
and defining w to be the element of I(d, 2d) associated to ∪j oddSwj,j+1 would give
an equivalent definition of Oπ.
7.2 Illustration by an example
We illustrate the map Oπ by means of an example. Let d = 15, and v =
(1, 2, 3, 4, 9, 10, 14, 16, 18, 19, 20, 23, 24, 25, 26). A monomial S in ON is shown
in Figure 7.2.1. Solid black dots indicate the elements that occur in S with non-
zero multiplicity. Integers written near the solid dots indicate multiplicities.
The O-depth of S is 5. The element (21, 9) has O-depth 3 although it has
depth 2 in S. Figure 7.2.2 shows the monomials S
1,2, S
3,4, and S
5,6. Solid
dots, open dots, and crosses indicate elements of these monomials respectively.
The monomial S is truly orthogonal at 1 and 3 but not at 5: σ1 = (28, 2),
σ3 = (21, 9), and σ5 = (15, 14).
Figure 7.2.3 shows the monomials S1,2, S3,4, and S5,6 of N and also their
decomposition into blocks, and Figure 7.2.4 the monomialsS′1,2, S
3,4, and S
We have
Sw = {(15, 14), (17, 16), (21, 10), (7, 4), (27, 24), (28, 3), (30, 1), (29, 2)}
hence w = (7, 9, 15, 17, 18, 19, 20, 21, 23, 25, 26, 27, 28, 29, 30). It is easy to check
that w ∈ I(d). The monomial S′ is the intersection with ON of the union of
S′1,2, S
3,4, and S
5,6—in other words it is just the monomial lying above d in
Figure 7.2.4.
diagonal
1         2        3        4       9       10      14     16     18      19     20     23     24      25     26  
2 1 3 4
3 2 6 2
2 3 2
3 1 2 3
Figure 7.2.1: The monomial S
diagonal
1         2        3        4       9       10      14     16     18      19     20     23     24      25     26  
2 1 3 4
3 2 6 2
2 1 3 2
3 1 2 3
Figure 7.2.2: S
1,2, S
3,4, and S
diagonal
1         2        3        4       9       10      14     16     18      19     20     23     24      25     26  
2 1 3 4
3 2 6 2
2 1 3 2
3 1 2
2 4 3 1 4 2 1 3 2
1 2 1 3 2 1
2 4 6 5 3
1 2 1 4
Figure 7.2.3: S1,2, S3,4, and S5,6
diagonal
1         2        3        4       9       10      14     16     18      19     20     23     24      25     26  
2 1 4
3 6 5 3
2 1 3 2 1
1 4 3 1 4 2 1 3 2
Figure 7.2.4: S′1,2, S
3,4, and S
7.3 A proposition about Sj,j+1
The aim of this subsection is to show that Sj,j+1 has depth no more than 2—
see item (1b) of Proposition 7.3.3. This basic fact was mentioned above in
the description of Oπ and is necessary (psychologically although not logically)
to make sense of the definitions of Sj and Sj+1. We prepare the way for
Proposition 7.3.3 by way of two preliminary propositions. The first of these is
about elements of O-depth j and j + 1 in S, the second about the relation of
these elements with σj .
Proposition 7.3.1 1. S
has no comparable elements.
2. For j an odd integer and β an element of S
j+1, there exists α in S
j such
that α > β. In particular, the row index of σj+1 (if σj+1 exists) is less
than the row index of σj.
Proof: (1) follows from Corollary 6.1.3 (4); (2) follows from Proposition 6.3.3
and Lemma 6.3.1 (5). �
Proposition 7.3.2 Let j be an odd integer and let S be truly orthogonal at j.
1. pv(σj) > ph(σj); if α > pv(σj), then α > σj; if α > σj , then α > ph(σj).
2. No element of S
j is comparable to pv(σj) or ph(σj).
3. No element of S
j+1 is comparable to ph(σj).
4. The following is not possible: α ∈ S
j , β ∈ S
j+1, and ph(α) > β.
Proof: (1) is trivial. (2) follows immediately from the definition of σj . We
now prove (3). First suppose β > ph(σj) for some β in S
j+1. By (2) of
Proposition 7.3.1, there exists α in S
j such that α > β. But then the row
index of α exceeds that of σj , a contradiction to the choice of σj .
We claim that it is not possible for β ∈ S
j+1 to satisfy ph(σj) > β. This
being a special case of (4), we need only prove that statement. So suppose that
α belongs to S
j and that ph(α) > β. Let C be a v-chain in S with tail α such
that O-depthC(α) = j (see Proposition 6.1.3 (1)). Concatenate C with α > β
and call the resulting v-chain D. Then, by Lemma 6.3.1 (4), α is of type H
in D, so that, by Lemma 6.3.1 (1), we have O-depthD(β) = O-depthD(α) + 2.
But, by Proposition 6.1.1 (2), O-depthD(α) = O-depthC(α) = j, so that
O-depth
(β) ≥ j + 2, a contradiction. �
Let Sj,j+1(ext) denote the set—not multiset—defined by:
Sj,j+1(ext) :=
Sj,j+1 ∪ {σj , σ
j } if S is truly orthogonal at j
Sj,j+1 otherwise
Here Sj,j+1 on the right stands for the underlying set of the multiset Sj,j+1
defined above. The set Sj,j+1(ext) is the disjoint union of the sets Sj(ext)
and Sj+1(ext) defined as follows (here again the terms on the right hand side
denote the underlying sets of the corresponding multisets):
Sj(ext) :=
∪ {pv(σ)} if S is truly orthogonal at j
otherwise
Sj+1(ext) :=
j+1 ∪
∪ {ph(σ)} if S is truly orthogonal at j
j+1 ∪
otherwise
Proposition 7.3.3 1. Sj(ext) (respectively Sj+1(ext)) is precisely the set
of elements of depth 1 (respectively 2) in Sj,j+1(ext). In particular,
(a) Neither Sj(ext) nor Sj+1(ext) contains comparable elements.
(b) The length of a v-chain in Sj,j+1(ext) is at most 2.
(c) There is a v-chain of length 2 in Sj,j+1 unless Sj+1(ext) is empty.
2. Let k be a positive integer, not necessarily odd. If there is in S an element
of O-depth at least k, then Sk(ext) is non-empty. The converse also holds
except possibly if k is even and S is truly orthogonal at k−1. In particular,
if Sk(ext) is non-empty, then there is an element of O-depth at least k−1.
Proof: (1): It is enough to show that every element of Sj(ext)(up) (respec-
tively Sj+1(ext)(up)) is of depth 1 (respectively 2) in Sj,j+1(ext)(up), for
• α > β implies α(up) > β(up) for elements α, β of N.
• Sj,j+1(ext) = Sj(ext) ∪Sj+1(ext).
• Sj,j+1(ext), Sj(ext), and Sj+1(ext) are symmetric.
In turn, it is enough to show the following:
(i) Every element of Sj(ext)(up) has depth 1.
(ii) Sj+1(ext)(up) has no comparable elements.
(iii) Every element of Sj+1(ext)(up) has depth at least 2.
Item (i) follows from Proposition 7.3.1 and Proposition 7.3.2 (2); item (ii)
from Proposition 7.3.1 (1) and Proposition 7.3.2 (3); item (iii) from Propo-
sition 7.3.1 (2) and Proposition 7.3.2 (1).
(2): The first assertion follows from Lemma 6.3.1 (5): if k is odd there is an
element ofO-depth k inS; if k is even and there is no element ofO-depth k inS,
then there is in S an element of O-depth k− 1 and of type H, so S is truly or-
thogonal at k−1. The second assertion is clear from the definition of Sk(ext). �
Corollary 7.3.4 1. No element of Sj,j+1 has depth more than 2.
2. Sj+1(ext) = Sj+1 and Sj(ext) ∩ Sj,j+1 = Sj (as sets). In particular,
Sj+1 = Sj,j+1∩Sj+1(ext) and Sj = Sj,j+1∩Sj(ext) as multisets defined
by the intersection of a multiset with a subset.
Proof: (1): Since Sj,j+1 ⊆ Sj,j+1(ext) (as sets), this follows immediately
from (1b) of the proposition above.
(2): Since the union of Sj+1(ext) (which always is contained in Sj,j+1) and
Sj(ext) ∩Sj,j+1 is all of Sj,j+1, and since Sj , Sj+1 are disjoint, it it enough
to show that Sj+1(ext) ⊆ Sj+1 and Sj(ext) ∩Sj,j+1 ⊆ Sj .
Now, since elements of Sj(ext) have depth 1 even in Sj,j+1(ext) (by item (1)
of the proposition above), it is immediate that Sj(ext) ∩ Sj,j+1 ⊆ Sj . And
it follows from the proof of item (iii) in the proof of item (1) of the proposi-
tion above that an element of Sj+1(ext) has depth 2 even in Sj,j+1 (not just
in Sj,j+1(ext)), so that Sj+1(ext) ⊆ Sj+1. �
7.4 Proof of Proposition 7.1.1
(1) The monomials Sj,j+1 are clearly symmetric. Observe that α in Sj,j+1 has
the same depth as α#, for α1 > α2 implies α(up) > α2(up) and α(down) >
α2(down) for α1, α2 in N. Thus the monomials Sk are symmetric. Since the
map π of [7] respects #—see Proposition 5.7 of [4]—it follows that Sw(k) and
S′k are symmetric. Therefore so are ∪kSw(k) and ∪kS
(2) This follows from Corollary 9.3.6.
(3) IfS is truly orthogonal at j, then pv(σj) and ph(σj) are diagonal elements
respectively in Sj and Sj+1—see Corollary 7.3.4 (2). Thus both Sj and Sj+1
have diagonal blocks in the sense of Proposition 5.10 (A) of [4]. It follows from
the result just quoted that both Sw(j) and Sw(j+1) meet the diagonal. It is
of course clear that each Sw(k) meets the diagonal at most once since diagonal
elements are clearly comparable but elements of Sw(k) are not by Lemma 4.9
of [7].
Suppose that S is not truly orthogonal at j. Then σj and σ
j belong to
different blocks—this is equivalent to the definition of S being not truly orthog-
onal at j. By Proposition 7.3.1 (2), it follows that σj+1 and σ
j+1 also belong
to different blocks. So neither Sj nor Sj+1 has a diagonal block.
(4) If S is not truly orthogonal at j, then neither Sj nor Sj+1 has a diagonal
block (as has just been said above), and it follows from Proposition 5.10 (A)
of [4] that neither S′j nor S
j+1 meets the diagonal.
So suppose that S is truly orthogonal at j. Then both Sj and Sj+1 have
a diagonal entry each of multiplicity 1, namely pv(σj) and ph(σj) respectively.
It is clear from the definition of σj that no element of Sj(up) shares its row
index with pv(σj). And it follows from Proposition 7.3.1 (2) that no element
of Sj+1(up) shares its row index with ph(σj). It now follows from the proof of
Proposition 5.10 (B) of [4]—see the last line of that proof—that neither S′j nor
S′j+1 meets the diagonal. �
7.5 More observations
Proposition 7.5.1 The length of any v-chain in Sj,j+1∪S
j+1 is at most 2.
Proof: By Corollary 7.3.4 (1), the length of any v-chain in Sj,j+1 is at most 2.
Applying Lemma 9.1.1 to Sj,j+1, we get the desired result. �
Proposition 7.5.2 1. For an element α′ = (r, c) of S′k(up), there exists an
element α = (r, C) of S
with C ≤ c.
2. For an element α′ = (r, c) of S′j+1(up), there exists an element α = (R, c)
of Sj+1(up) with r ≤ R.
3. For an element α′ of S′j+1(up), there exists an element α of S
j with
α > α′.
Proof: (1) That there exists α inSk(up) with C ≤ c follows from the definition
of S′k(up). Clearly such an α cannot be on the diagonal, so α belongs to S
(2) As in the proof of (1), it follows from the definition of S′j+1 that there
exists α = (R, c) in Sj+1 with r ≤ R. If α lies strictly below the diagonal, then
c > R∗, so that α∗ = (c∗, R∗) > α′ = (r, c), a contradiction to Lemma 9.1.1 (α∗
belongs to Sj+1 by the symmetry of Sj+1). Thus α belongs to Sj+1(up).
(3) Writing α′ = (r, c), by (1), we can find an β = (r, C) in S
j+1 with C ≤ c.
By Proposition 7.3.1 (2), there exists α in S
j,j+1 such that α > β. �
Corollary 7.5.3 If in S′j(up)∪S
j+1(up) there exists an element with horizon-
tal projection in N, then S is truly orthogonal at j.
Proof: Follows directly from Proposition 7.5.2 (1) and (3). �
Proposition 7.5.4 The O-depth of an element in S
j ∪ S
j+1 is at most 2.
More strongly, the O-depth of an element in S
j,j+1 ∪S
j(up) ∪S
j+1(up) is at
most 2.
Proof: It is enough to show that no element in S′j(up)∪S
j+1(up) hasO-depth
more than 2, for we may assume by increasing multiplicities that S
j ⊆ S
j(up)
and S
j+1 ⊆ S
j+1(up) (as sets). It follows from Proposition 7.5.1 that a v-chain
in S′j(up) ∪ S
j+1(up) has length at most 2. Let α
1 = (r1, c1) > α
2 = (r2, c2)
be such a v-chain. It follows from the proof of Corollary 4.14 (2) of [7] that
α′1 ∈ S
j(up) and α
2 ∈ S
j+1(up). By item (1) of Lemma 6.3.1, it is enough to
rule out the following possibility: α′1 is of type H in α
1 > α
2 and ph(α
1) > α
Suppose that this is the case. By Proposition 7.5.2 (1) and (2), it follows
that there exist elements α1 = (r1, C1) ∈ S
j,j+1 and α2 = (R2, c2) ∈ Sj+1(up)
with C1 ≤ c1 and r2 ≤ R2. Since ph(α
1) > α
2, it follows that α1 > α2. Now, if
α2 = ph(σj), then Proposition 7.3.2 (2) is contradicted; if α2 belongs to S
Proposition 7.3.2 (4) is contradicted (because ph(α1) > α2). �
8 The map Oφ
The purpose of this section is to describe the map Oφ and prove some basic
facts about it. Certain proofs here refer to results from §9, but there is no
circularity—to postpone the definition of Oφ until all the results needed for it
have been proved would hurt rather than help readability. As in §7, the symbol j
will be reserved for an odd integer throughout this section.
8.1 Description of Oφ
The map Oφ takes as input a pair (w,T), where T is a monomial, possibly
empty, in ON and w ≥ v an element of I(d) that O-dominates T, and produces
as output a monomial T∗ of ON. To describe Oφ, we first partition T into
subsets Tw,j,j+1. As the subscript w in Tw,j,j+1 suggests, this partition depends
on w.
For an odd integer j, let Sjw (respectively Sw,j,j+1) denote the subset of Sw
consisting of those elements that are j-deep (respectively that are j deep but
not j + 2 deep, or equivalently of depth j or j + 1) in Sw in the sense of [7,
§4]. Since Sw is distinguished, symmetric, and has evenly many elements on
the diagonal d, it follows that Sjw and Sw,j,j+1 too have these properties, and
that, in fact, the number of diagonal elements of Sw,j,j+1 is either 0 or 2 (in
the latter case, the elements have to be distinct since Sw is distinguished and
so is multiplicity free). Let us denote by wj and wj,j+1 the elements of I(d)
corresponding to Sjw and Sw,j,j+1 by Proposition 5.2.1.
Let Tw,j,j+1 denote the subset of T consisting of those elements α such that
• every v-chain in T with head α is O-dominated by wj , and
• there exists a v-chain in T with head α that is not O-dominated by wj+2.
It is evident that the subsets Tw,j,j+1 are disjoint (as j varies over the odd
integers) and that their union is all of T (for w = w1 O-dominates all v-chains
in T by hypothesis and Sjw is empty for large j and so w
j = v). In other words,
the Tw,j,j+1 form a partition of T.
Lemma 8.1.1 1. The length of a v-chain in Tw,j,j+1∪T
w,j,j+1 is at most 2.
In fact, the O-depth of any element in Tw,j,j+1 is at most 2.
2. wj,j+1 O-dominates Tw,j,j+1.
Proof: The lemma follows rather easily from Corollary 9.2.3 as we now show.
Let C be a v-chain in Tw,j,j+1. Let τ be the tail of C. Choose a v-chain D
in T with head τ that is not O-dominated by wj+2. Let E be the concate-
nation of C with D. Since the head of E belongs to Tw,j,j+1, it follows that
E is O-dominated by wj . It follows from (the only if part of) Corollary 9.2.3
(applied with S = E and x = wj) that wj,j+1 O-dominates E
1 ∪ E
2 and
wj+2 O-dominates E3,pr. This means τ 6∈ E3,pr, so τ ∈ E
1 ∪ E
2 , and so
C ⊆ E
1 ∪ E
2 . This proves (2). By Proposition 6.1.1 (2), the O-depths of
elements of C are the same in C and E, so C ⊆ C
1 ∪ C
2 , which proves the
second assertion of (1). The first assertion of (1) follows from the second (see
Lemma 6.2.1). �
Corollary 8.1.2 wj,j+1 dominates Tw,j,j+1 ∪ T
w,j,j+1 in the sense of [7].
Proof: This follows from (2) of Lemma 8.1.1 and Corollary 5.3.6 (the latter
applied with S = Tw,j,j+1 and w = wj,j+1). �
We may therefore apply the map φ of [7, §4] to the pair (wj,j+1,Tw,j,j+1 ∪
w,j,j+1) to obtain a monomial (Tw,j,j+1∪T
w,j,j+1)
⋆ in N. In applying φ, there
is the partitioning of Tw,j,j+1 ∪ T
w,j,j+1 into “pieces”, these being indexed by
elements of Swj,j+1 = Sw,j,j+1—observe that the elements of depth 1 (respec-
tively 2) of Sw,j,j+1 are precisely those of Sw of depth j (respectively j + 1).
We denote by Pβ the piece of Tw,j,j+1∪T
w,j,j+1 corresponding to β in Swj,j+1 .
We also use the notation P∗β as in [7]. Moreover, we will use the phrase piece
of T (with respect to w being implicitly understood) to refer to a piece of
Tw,j,j+1 ∪ T
w,j,j+1 for some odd integer j.
Caution: Thinking of T as a monomial in N and w as an element of I(d, 2d) that
dominates it, there is, as in [7], the notion of “piece of T” (with respect to w).
The two notions of “piece” are different.
Lemma 8.1.3 1. The monomial (Tw,j,j+1∪T
w,j,j+1)
⋆ is symmetric and has
either none or two distinct diagonal elements depending exactly on whether
Swj,j+1 = Sw,j,j+1 has 0 or 2 elements on the diagonal.
2. The depth of (Tw,j,j+1 ∪ T
w,j,j+1)
⋆ is 2; and ∪β∈(Sw)jP
β, ∪β∈(Sw)j+1P
are respectively the elements of depth 1 and 2 in (Tw,j,j+1 ∪ T
w,j,j+1)
Proof: (1) The symmetry follows by combining Proposition 5.6 of [4], which
says that the map π respects the involution #, with Proposition 4.2 of [7], which
says that π and φ are are inverses of each other.
The assertion about diagonal elements follows by combining item (B) of [4,
Proposition 5.10], which is an assertion about the existence and relative mul-
tiplicities of diagonal elements in B and B′ where B is a diagonal block of a
monomial in N, and Proposition 4.2 of [7].
(2) It follows from Propositions 4.2 of [7] that the map π (described in §4 of
that paper) applied to (Tw,j,j+1∪T
w,j,j+1)
⋆ results in the pair (wj,j+1,Tw,j,j+1∪
Tw,j,j+1). It now follows from Lemma 4.16 of [7] that the depth of (Tw,j,j+1 ∪
w,j,j+1)
⋆ is exactly 2. The latter assertions again follow from the results of [7]—
in fact, the proof that π ◦ φ is identity on pages 47–49 of [7] shows that the P∗β
are the blocks in the sense of [7] of the monomial (Tw,j,j+1 ∪ T
w,j,j+1)
Suppose that (Tw,j,j+1 ∪ T
w,j,j+1)
⋆ contains the pair (a, a∗), (b, b∗) of di-
agonal elements with a > b. We call the pair (b, a∗), (a, b∗) the “twists,” and
set δj := (b, a
∗). In other words, δj is the element of the twisted pair that lies
above the diagonal—observe that the twisted elements are reflections of each
other. We allow ourselves the following ways of expressing the condition that
(Tw,j,j+1 ∪ T
w,j,j+1)
⋆ has diagonal elements: δj exists; w is diagonal at j (the
latter expression is justified by the lemma above).
With notation as above, consider the new monomial defined as
(Tw,j,j+1 ∪ T
w,j,j+1)
⋆ if w is not diagonal at j(
(Tw,j,j+1 ∪ T
w,j,j+1)
⋆ \ d
∪ {δj, δ
j } if w is diagonal at j
This new monomial is symmetric and contains no diagonal elements. Its inter-
section with ON is denoted T⋆w,j,j+1. In other words, T
w,j,j+1 is the intersection
of the new monomial with the subset of N of those elements that lie strictly
above the diagonal.
The union of T⋆w,j,j+1 over all odd integers j is defined to be T
w, the result
of Oφ applied to (w,T). This finishes the description of the map Oφ.
For β in Sw,j,j+1(up), we define the “orthogonal piece-star” OP
β corre-
sponding to β as
OP∗β :=
P∗β = P
β(up) if β is not on the diagonal
P∗β ∩ON if β ∈ (Sw)j+1 is on the diagonal
{P∗β ∩ON} ∪ {δj} if β ∈ (Sw)j is on the diagonal
(8.1.1)
With this, we can say that T⋆w is the union of OP
β as β varies over Sw(up).
Lemma 8.1.4 Suppose that (Tw,j,j+1 ∪ T
w,j,j+1)
⋆ contains the pair (a, a∗),
(b, b∗) of diagonal elements with a > b. Let
. . . , (r1, c1), (a, a
∗), (c∗1, r
1), . . . ; . . . , (r2, c2), (b, b
∗), (c∗2, r
2), . . .
be respectively the elements of depth 1 and 2 of (Tw,j,j+1 ∪ T
w,j,j+1)
⋆ arranged
in increasing order of row and column indices. Then
1. c1 ≤ a
∗ and r1 ≤ b (assuming (r1, c1) exists); and
2. r2 < b and c2 ≤ b
∗ (assuming (r2, c2) exists).
Proof: (1) Suppose that (r1, c1) exists. It is clear that c1 ≤ a
∗. From way the
map φ of [7] is defined, it follows that (r1, a
∗) is an element of Tw,j,j+1. Suppose
that r1 > b. Then ph(r1, a
∗) = (r1, r
1) belongs to N. We consider two cases.
If (r2, c2) exists, then, again from the definition of the map φ, it follows that
(r2, b
∗) is an element of Tw,j,j+1. But then ph(r1, a
∗) = (r1, r
1) > (b, b
∗) and
(b, b∗) dominates (r2, b
∗), which means that the v-chain (r1, a
∗) > (r2, b
∗) (note
that a∗ < b∗ because a > b by hypothesis) in Tw,j,j+1 has O-depth more than 2,
a contradiction to Lemma 8.1.1 (1).
Now suppose that (r2, c2) does not exist. (Then (b, b
∗) is the diagonal ele-
ment in (Sw)j+1.) Consider the singleton v-chain C := {(r1, a
∗)} in Tw,j,j+1.
Then SC = {(a, a
∗), (r1, r
1)} which is not dominated by wj,j+1, a contradiction
to Lemma 8.1.1 (2).
(2) Suppose that (r2, c2) exists. Then there exists, by the definition of the
map φ, an element (r2, b
∗) in Tw,j,j+1. Since (r2, b
∗) lies above the diagonal, it
follows that r2 < b. That c2 ≤ b
∗ is clear. �
8.2 Basic facts about Tw,j,j+1 and T
w,j,j+1
Lemma 8.2.1 1. Let α′ > α be elements of T. Let j and j′ be the odd
integers such that α′ ∈ Tw,j′,j′+1 and α ∈ Tw,j,j+1. Then j
′ ≤ j.
2. If, further, either
(a) there exists µ in T such that α′ > µ > α, or
(b) α′ ∈ Pβ′ for β
′ in (Sw)j′+1,
then j′ < j.
Proof: (1) By hypothesis, every v-chain with head α′ is O-dominated by wj
This implies, by Corollary 6.1.2, that every v-chain with head α is O-dominated
by wj
. This shows j′ ≤ j.
(2a) Suppose that j′ = j. It follows from (1) that α′, µ, and α all belong
to Tw,j,j+1. But then α
′ > µ > α is a v-chain of length 3 in Tw,j,j+1, a
contradiction to Lemma 8.1.1 (1).
(2b) Suppose that j′ = j. Then α′ > α is a v-chain in Tw,j,j+1. Being of
length 2, it cannot be dominated by (Sw)j+1, which means, by the definition
of Pβ′ , that α
′ cannot belong to Pβ′ , a contradiction. �
Proposition 8.2.2 1. The length of a v-chain in T⋆w,j,j+1 is at most 2.
2. The O-depth of T⋆w,j,j+1 is at most 2.
3. ∪β∈(Sw)j(up)OP
β is precisely the set of depth 1 elements of T
w,j,j+1 (in
particular, no two elements there are comparable); if δj exists, then it is
the last element of ∪β∈(Sw)j(up)OP
β when the elements are arranged in
increasing order of row and column indices.
4. ∪β∈(Sw)j+1(up)OP
β is precisely the set of depth 2 elements of T
w,j,j+1 (in
particular, no two elements there are comparable); if δj exists, then its
row index exceeds the row index of any element in ∪β∈(Sw)j+1(up)OP
Proof: For (1), it is enough, given Lemma 8.1.3 (2), to show that δj is not
comparable to any element of depth 1 of (Tw,j,j+1 ∪T
w,j,j+1)
⋆, and this follows
from Lemma 8.1.4 (1). In fact, the above argument proves also (3).
For (4), it is enough, given Lemma 8.1.3 (2), the symmetry of the monomi-
als involved in that lemma, and the observation that α > β implies α(up) >
β(up) for elements α, β of N, to show the following: if (a, a∗) > γ = (e, f)
for γ an element of (Tw,j,j+1 ∪ T
w,j,j+1)
⋆ lying (strictly) above the diagonal,
then δj > γ. But this follows from Lemma 8.1.4 (2): γ is a depth 2 ele-
ment in (Tw,j,j+1 ∪ T
w,j,j+1)
⋆, and we have e ≤ r2 < b (and a
∗ < f since
(a, a∗) > γ). In fact, the above argument proves also (2): observe that f ≤ b∗
(Lemma 8.1.4 (2)). �
9 Some Lemmas
The main combinatorial results of this paper are Propositions 4.1.1 and 4.1.2.
They are analogues respectively of Propositions 4.1 and 4.2 of [7]. We have
tried to preserve the structure of the proofs in [7] of those propositions. The
proofs in [7] rely on certain lemmas and it is natural therefore to first establish
the orthogonal analogues of those. The purpose of this section is precisely that.
Needless to say that the lemmas (especially those in §9.4) may be unintelligible
until one tries to read §10.
The division of this section into four subsections is also suggested by the
structure of the proofs in [7]. Each subsection has at its beginning a brief
description of its contents.
9.1 Lemmas from the Grassmannian case
In this subsection, the terminology and notation of [7, §4] are in force. The state-
ments here could have been made in [7, §4] and would perhaps have improved
the efficiency of the proofs there, but do not appear there explicitly.
Let S be a monomial in N. Recall from [7] the notion of depth of an ele-
ment α in S: it is the largest possible length of a v-chain in S with tail α and
denoted depth
α. The depth of S is the maximum of the depths in it of all its
elements. We denote by Sk the set of elements of depth k of S (as in [7]) and
by Sk the set of elements of depth at least k of S.
Caution: For a monomial S of ON, we have introduced in §7.1 the notation Sk.
That is different from the Sk we have just defined.
Lemma 9.1.1 Let S be a monomial in N, and let π(S) = (w,S′), where π is
the map defined in [7, §4]. Then the maximum length of a v-chain in S∪S′ is
the same as the maximum length of a v-chain in S.
Proof: We use the notation of [7, §4] freely. Let d be the maximum length of
a v-chain in S. Suppose α1 > . . . > αℓ is a v-chain in S ∪S
′. Let i1, . . . , iℓ be
such that αj belongs to Sij ∪ S
(the integers ij are uniquely determined—
see Corollary 5.4 of [4]). We claim that i1 < . . . < iℓ. This suffices to prove the
lemma, for Sk ∪S
k is empty for k > d.
To prove the claim, it is enough to show i1 < i2. It follows from Lemma 4.10
of [7] that i1 6= i2. We now assume that i1 > i2 and arrive at a contradiction.
First suppose that α1 ∈ Si1 . Then, by the definition ofSi1 , there exists β inSi2
with β > α1. Now β > α2 and both β, α2 belong to Si2 ∪S
, a contradiction
to [7, Lemma 4.10]. If α1 = (r, c) belongs to S
, then, by the definition of S′i1 ,
there exists (r, a) in Si1 with a ≤ c, and there exists β in Si2 with β > (r, a).
This leads to the same contradiction as before. �
Lemma 9.1.2 Let B and U be monomials in N. Assume that
• the elements of B form a single block (in the sense of [7, Page 38]).
• U has depth 1 (equivalently, there are no comparable elements in U).
• for every β = (r, c) in B, there exist γ1(β) = (R1, C1), and γ2(β) =
(R2, C2) in U such that
C1 < c, C2 < R1, r < R2
(this holds, for example, when there exists γ(β) in U such that γ(β) > β:
take γ1(β) = γ2(β) = γ(β)).
Then there exists a unique block C of U such that w(C) > w(B).
Proof: It is useful to isolate the following observation:
Lemma 9.1.3 Let (r1, c1) and (r2, c2) be elements of N with c2 < r1 ≤ r2. Let
γ11 = (R
1 ), γ
1 = (R
1 ) and γ
2 = (R
2 ), γ
2 = (R
2 ) be elements
of N such that
1. C11 ≤ c1, C
1 < R
1, r1 ≤ R
2. C12 ≤ c2, C
2 < R
2, r2 ≤ R
3. No two of γ11 , γ
1 , γ
2 , γ
2 are comparable (they could well be equal and this
is important for us—see our definition of comparability).
Then the monomial {γ11 , γ
1 , γ
2 , γ
2} consists of a single block.
Proof: It follows from assumption (1) that γ11 and γ
1 belong to a single block:
• if R11 < R
1, then C
1 < R
1 becomes relevant;
• if R21 < R
1, then the other two inequalities in (1) become relevant:
C11 ≤ c1 < r1 ≤ R
Similarly it follows from assumption (2) that γ12 and γ
2 belong to a single block.
We therefore need only consider the cases when, in the arrangement of the
elements {γ11 , γ
1 , γ
2 , γ
2} in increasing order of row indices, both γ
1 , γ
1 come
before or after γ12 , γ
2 . In the former case, the first sequence of inequalities
below shows that γ21 and γ
2 belong to the same block, and we are done; in
the latter case, the second sequence of inequalities below shows that γ22 and γ
belong to the same block, and we are done:
• C12 ≤ c2 < r1 ≤ R
• C11 ≤ c1 < r1 ≤ r2 ≤ R
Continuing with the proof of Lemma 9.1.2, we first prove the existence part.
Arrange the elements of B in non-decreasing order of row numbers as well as
column numbers (this is possible since there are no comparable elements in B).
If β1 = (r1, c1) and β2 = (r2, c2) are successive elements, then c2 < r1 ≤ r2
(since B is a single block). Apply Lemma 9.1.3 with γ11 = γ
1(β1), γ
1 = γ
2(β1),
and γ12 = γ
1(β2), γ
2 = γ
1(β2). We conclude that {γ
1 , γ
1 , γ
2 , γ
2} belongs to a
single block, say C, of U. Continuing thus, we conclude that all γ1(β) and γ2(β),
as β varies over B, belong to C. Since the row (respectively column) index of
w(C) is the maximum (respectively minimum) of all row (respectively column)
indices of elements of C (and similarly for B), it follows that w(C) > w(B).
To prove uniqueness, let C1 and C2 be two blocks of U with w(C1) > w(B)
and w(C2) > w(B). Apply the lemma with (r1, c1) = (r2, c2) = w(B) and
γ11 = γ
1 = w(C1) and γ
2 = γ
2 = w(C2); it follows from [7, Lemma 4.9] that
w(C1) and w(C2) are not comparable. But, unless C1 = C2, neither is the mono-
mial {w(C1), w(C2)} a single block, again by [7, Lemma 4.9]. �
Lemma 9.1.4 Let S be a monomial in N and x an element of I(d, n). For
x to dominate S it is necessary and sufficient that for every α = (r, c) in S
there exist β = (R,C) in Sx with C ≤ c, r ≤ R, and depthSxβ ≥ depthSα.
(Here Sx denotes the distinguished monomial in N associated to x as in [7,
Proposition 4.3].)
Proof: The lemma is a corollary of [7, Lemma 4.5] as we now show.
First suppose that x dominatesS. Let α = (r, c) be an element ofS, and C a
v-chain in S with tail α and length depth
α. Since x dominates C, there exists,
by [7, Lemma 4.5], a chain in D in Sx of length depthSα and tail β = (R,C)
with C ≤ c and r ≤ R, and we are done with the proof of the necessity.
To prove the sufficiency, let C : α1 = (r1, c1) > . . . > αk = (rk, ck) be a
v-chain in S. By hypothesis, there exist β1 = (R1, C1), . . . , βk = (Rk, Ck) in
Sx with Ci ≤ ci, ri ≤ Ri, and depthSxβi = i for 1 ≤ i ≤ k (observe that
replacing the ≥ in the latter condition of the statement by an equality yields
an equivalent statement). We claim that β1 > . . . > βk. By [7, Lemma 4.5], it
suffices to prove the claim.
Since βk has depth k in Sx, there exists a β
k−1 = (R
k−1, C
k−1) of depth
k − 1 in Sx such that β
k−1 > βk. It follows from the distinguishedness of Sx
that that β′k−1 = βk−1: if not, then we have two distinct elements of the same
depth (namely k−1) in Sx both dominating αk, a contradiction. So βk−1 > βk,
and the claim is proved by continuing in a similar fashion. �
Let x be an element of I(d, n). Let Sx denote the distinguished monomial
in N associated to x as in [7, Proposition 4.3]. For k a positive integer, let xk
denote the element of I(d, n) corresponding to the distinguished subset (Sx)k.
For a monomial S ofN, let Sk,k+1 := Sk∪Sk+1. Let xk,k+1 denote the element
of I(d, n) corresponding to the distinguished monomial (Sx)k,k+1; let x
k denote
the element of I(d, n) corresponding to the distinguished subset (Sx)
Caution: For a monomial S of ON and an odd integer j, we have introduced
in §7.1 the notation Sj,j+1. That is different from the Sk,k+1 just defined.
Corollary 9.1.5 x dominates S ⇔ xk dominates Sk ∀ k ⇔ x1,2 dominates
S1,2 and x
3 dominates S3.
Proof: The first equivalence is a restatement of the lemma: in the statement
of the lemma we could equally well have written depth
β = depth
α. The
second follows from the first and the following observations: (S1,2)1 = S1,
(S1,2)2 = S2, (S
3)k = Sk+2; and (x1,2)1 = x1, (x1,2)2 = x2, (x
3)k = xk+2. �
9.2 Orthogonal analogues of Lemmas of 9.1
Lemma 9.2.2 below is the orthogonal analogue of Lemma 9.1.4 (more precisely,
that of the first assertion of Corollary 9.1.5). The following proposition will be
used in its proof.
Proposition 9.2.1 Let x be an element of I(d) and S a monomial in ON.
Then x O-dominates S
1 ∪ S
2 if and only if it O-dominates every v-chain
in S of O-depth at most 2.
Proof: The “if” part is immediate from definitions (in any case, see also
Proposition 7.5.4). For the “only if” part, let C be a v-chain in S of O-depth
at most 2. Our goal is to show that x dominates SC . For this, it is enough, by
Corollary 9.1.5, to show that x1 dominates (SC)1 and x2 dominates (SC)2 (by
choice of C, (SC)k is empty for k ≥ 3).
Let α′ ∈ (SC)1. Choose α in C such that α
′ ∈ SC,α. Choose α0 in S
1 such
that α0 dominates α. Since x O-dominates the singleton v-chain {α0}, it follows
that x1 dominates q{α0},α0 . We claim that q{α0},α0 dominates α
′. To prove the
claim, we need only rule out the possibility that α0 is of type S in {α0} and α
of type V in C. Since α′ ∈ (SC)1, it follows from Proposition 5.3.4 (1) that α
is the first element of C. In particular, if α is of type V in C, then ph(α) ∈ N,
so ph(α0) ∈ N, and α0 is of type H in {α0}. The claim is thus proved.
Now consider an element of (SC)2. Observe that the length of C is at most 2
(Lemma 6.2.1). So our element is either the horizontal projection ph(α) of the
head α of C, or it is qC,β where β is the tail of C. In the first case, let α0 be
as in the previous paragraph, and proceed similarly. It is clear that ph(α0) ∈ N
(because ph(α) ∈ N); x2 dominates ph(α0) and so also ph(α).
Now we handle the second case. If β ∈ S
2 , then C is contained in S
and there is nothing to prove. So assume that O-depth
(β) ≥ 3. Choose a
v-chain D in S with tail β, O-depthD(β) ≥ 3, and with the good property
as in Proposition 6.3.3. There occurs in D an element of O-depth 3, say δ.
(Lemma 6.3.1 (5)). Let A denote the part δ > . . . of D and C′ the part up to
but not including δ. There clearly is an element—call it µ—of depth 2 in SD
that dominates qD,β. This element µ belongs to SC′ (Corollary 6.3.5 (3)). Since
D has the good property of Proposition 6.3.3, C′ ⊆ S
2 , so µ is dominated
by an element in (Sx)2. In particular, qD,β is dominated by the same element
of (Sx)2.
We are still not done, for it is possible that qD,β be β and qC,β be pv(β).
Suppose that this is the case. Then α > β is connected. So ph(α) ∈ N and the
legs of α and β intertwine. As seen above in the third paragraph of the present
proof, there is an element of (Sx)2 that dominates ph(α). By the distinguished-
ness of Sx, it follows that the element in (Sx)2 dominating β is the same as
the one dominating ph(α). By the symmetry of Sx, this element lies on the
diagonal and so dominates pv(β), and, finally, we are done with the proof in the
second case. �
Lemma 9.2.2 Let S be a monomial in ON and x an element of I(d). For x to
O-dominate S it is necessary and sufficient that, for every odd integer j, every
v-chain in S
j+1 is O-dominated by xj,j+1.
Proof: First suppose that x dominates S. Let j be an odd integer and let A
a v-chain in S
j+1. We need to show that xj,j+1 dominates SA. For this,
we may assume that A is maximal (by Corollary 6.1.2). By Corollary 6.1.3 (3),
the length of A is at most 2. By Lemma 6.3.1 (5) (b), for every β in S
j+1 there
exists α in S
j with α > β. Thus we may assume that the head α of A belongs
It is enough to show (see [7, Lemma 4.5]) that for any v-chain E in SA
• the length of E is at most 2;
• there exists an x-dominated monomial in N containing E and the head
of E is an element of depth at least j in that monomial.
The first of these conditions holds by Proposition 7.5.4. We now show that the
second holds.
We may assume that E is maximal in SA. By Proposition 5.3.4 (1), the
head of E is qA,α. Let C a v-chain in S with tail α such that O-depthC(α) = j.
Let D be the concatenation of C with A. We claim that the monomial SD has
the desired properties. That SD is x-dominated is clear (since x O-dominates
S). By Corollary 6.3.5, it follows that qD,α = qA,α and SA ⊆ SD (in particular
that E ⊆ SD). By Proposition 6.1.1 (2), O-depthD(α) = O-depthC(α) = j,
that is, depth
qD,α = j. The proof of the necessity is thus complete.
To prove the sufficiency, proceed by induction on the largest odd integer J
such that S
J+1 is non-empty. When J = 1, there is nothing to prove,
for S
1 ∪ S
2 = S and x1,2 O-dominates S
1 ∪ S
2 . So suppose that J ≥
3. We implicitly use Corollary 6.3.7 in what follows. By induction, x3 O-
dominates S3,4.
Let D be a v-chain in S. Our goal is to show that x dominates SD. Let
α be the element of D with O-depthD(α) = 3—such an element exists, by
Lemma 6.3.1 (5) (if there exists in D an element of O-depth in D exceed-
ing 2); the following proof works also in the case when α does not exist. Let
A be the part α > . . . of D, and C′ the part up to but not including α. By
Proposition 6.1.1 (2), the O-depth (in C′) of elements of C′ is at most 2. By
Proposition 9.2.1, x1,2 dominates SC′ . By Corollary 6.3.5 (3), (SD)1,2 = SC′
and (SD)
3 = SA. Since A ⊆ S
3,4, it follows that x3 dominates SA (induction
hypothesis). Finally, by an application of Corollary 9.1.5, we conclude that x
dominates SD. �
Corollary 9.2.3 Let S be a a monomial in ON and x an element of I(d). For
x to O-dominate S it is necessary and sufficient that x1,2 O-dominate S
and x3 O-dominate S3,4.
Proof: It is easy to see that (x3)j,j+1 = xj+2,j+3; it follows from Proposi-
tion 6.3.6 that (S3,4)
j ∪ (S
j+1 = S
j+2 ∪S
j+3. The assertion follows from
the lemma. �
9.3 Orthogonal analogues of some lemmas in [7]
The proofs of Propositions 4.1 and 4.2 of [7] are based on assertion 4.9–4.16 (of
that paper). Assertion 4.9 being a statement about a single Sk, it is applicable
in the present situation. Since references to it are frequent, we recall it below
as Lemma 9.3.1. As to assertions 4.10–4.16 of [7], assertions 9.3.2, 9.3.4–9.3.9
below are their respective analogues.
A block of a monomial S in ON means a block of Sj,j+1 in the sense of [7]
for some odd integer j.
Caution: Considering S as a monomial in N, there is the notion of a “block”
of S as in [7], which has in fact been used in §9.1, and which is different from the
notion just defined. Both notions are used and it will be clear from the context
which is meant.
Throughout this section S denotes a monomial in ON and j an integer (not
necessarily odd).
Lemma 9.3.1 If B1, . . . ,Bl are the blocks in order from left to right of some
Sk, and w(B1) = (R1, C1), w(B2) = (R2, C2), . . ., w(Bl) = (Rl, Cl), then
C1 < R1 < C2 < R2 < . . . < Rl−1 < Cl < Rl
Proof: This is merely a recall Lemma 4.9 of [7]. In any case it follows easily
from the definitions. �
Lemma 9.3.2 No two elements of Sk(ext) ∪ S
k are comparable. More pre-
cisely, it is not possible to have elements α > β both belonging to Sk(ext)∪S
Proof: It follows from Lemma 9.3.1 that Sk ∪ S
k contains no comparable
elements. If k is even, then Sk(ext) = Sk (Corollary 7.3.4 (2)); if k is odd, we
may assumeSk(ext) = Sk (as sets) by increasing the multiplicity of σk in S
Lemma 9.3.3 For integers i ≤ k, there cannot exist γ ∈ S′i(up) and β ∈ S
such that β > γ. For integers i < k, there cannot exist γ ∈ S′i(up) and β ∈ S
such that β dominates γ.
Proof: Let γ ∈ S′i(up) and β ∈ S
. If i = k and β > γ, then we get a
contradiction immediately to Lemma 9.3.2. Now suppose that i < k and that
β dominates γ. Apply Corollary 6.3.4 (the notation of the corollary being sug-
gestive of how exactly to apply it). Let α be as in its conclusion. The chain
α > γ contradicts Lemma 9.3.2 in case i is odd and either Lemma 9.3.2 or
Proposition 7.5.4 in case i is even. �
Lemma 9.3.4 For (r, c) in S′, there exists a unique block B of S with (r, c)
in B′.
Proof: The existence is clear from the definition of S′. For the uniqueness,
suppose that B and C are two distinct blocks of S with (r, c) in both B′ and C′.
We will show that this leads to a contradiction.
Let i and k be such that B ⊆ Si and C ⊆ Sk. From Lemma 4.11 of [7] (of
which the present lemma is the orthogonal analogue) it follows that i 6= k, so
we can assume without loss of generality that i < k. By applying the involu-
tion # if necessary, we may assume that (r, c) ∈ S′i(up). Now there exists an
element (r, a) in C with a ≤ c (this follows from the definition of C′). Clearly
(r, a) ∈ S
. Taking β = (r, a) and γ = (r, c), we get a contradiction to
Lemma 9.3.3. �
Lemma 9.3.5 Let i < j be positive integers.
1. Given a block B of Sj, there exists a unique block C of Si such that
w(C) > w(B).
2. Given an element β in Sj(ext)∪S
j, there exists α in Si such that α > β.
Proof: (1): The assertion follows by applying Lemma 9.1.2 with B = B
and U = Si. We need to make sure however that the lemma can be applied.
More precisely, we need to check that for every β = (r, c) in B there exist
γ1(β) = (R1, C1) and γ2(β) = (R2, C2) in Si such that C
1 < c, C2 < R1, and
r < R2. We may assume β = β(up), for, if β = β(down), then β(up) also belongs
to Sj because Sj is symmetric, and we can set γ
1(β) = γ2(β(up))(down),
and γ2(β) = γ1(β(up))(down)—note that these two belong to Si since Si is
symmetric.
We consider three cases:
1. β belongs to S.
2. β = ph(σj−1) (in particular, j is even and S is truly orthogonal at j − 1).
3. β = pv(σj) (in particular, j is odd and S is truly orthogonal at j).
Define β′ to be β in case 1, σj−1 in case 2, and σj in case 3. Let C be a v-chain
in S with tail β′ and having the good property as in Proposition 6.3.3.
First suppose that there exists in C an element of O-depth i and denote it
by γ. If ph(γ) 6∈ N (this can happen only in case 1), then set γ
1(β) = γ2(β) = γ.
Now suppose ph(γ) ∈ N. Then γ ∈ Si except when γ = σi with i odd and σi
has multiplicity 1 in S. If γ ∈ Si, take γ
1(β) = γ and γ2(β) = γ# = γ(down);
if γ 6∈ Si, then take γ
1(β) = γ2(β) = pv(γ).
Now suppose that C has no element ofO-depth i. Then, by Lemma 6.3.1 (5),
i is even and there exists in C an element of O-depth i − 1. This element
of C is of type H by Lemma 6.3.1 (1), so S is truly orthogonal at i − 1. Set
γ1(β) = γ2(β) = ph(σi−1).
(2): This proof parallels the proof of (1) above. As in the above proof, we
may assume that β = β(up). Suppose β = (r, c) belongs to S′j. Then there
exists (r, a) ∈ Sj with a ≤ c. Since S
j does not meet the diagonal, it is clear
that (r, a) ∈ ON, and thus it is enough to prove the assertion for β ∈ Sj(ext).
So now take β ∈ Sj(ext). Let β
′ and C be in the proof of (1). First suppose
that there exists in C an element of O-depth i. Denote it by γ. If γ ∈ Si, then
take α = γ. If γ 6∈ Si, then pv(γ) ∈ Si, and we take α = pv(γ). In case there is
no element in C of O-depth i, we take α = ph(σi−1) (see the above proof). �
Corollary 9.3.6 If B and B1 are blocks of S with w(B) = (r, c) and w(B1) =
(r1, c1), then exactly one of the following holds:
c < r < c1 < r1, c1 < r1 < c < r,
c < c1 < r1 < r, or c1 < c < r < r1.
Proof: This is a formal consequence of Lemmas 9.3.1 and 9.3.5, just as Corol-
lary 4.13 of [7] is of Lemmas 4.9 and 4.12 of that paper. �
Corollary 9.3.7 If w(B) > w(C) for blocks B ⊆ Si and C ⊆ Sj of S, then
i < j.
Proof: This is a formal consequence of Lemmas 9.3.1 and 9.3.5. It follows
from the first lemma that i 6= j. Suppose i > j. Then there exists by the second
lemma a block C′ ⊆ Sj such that w(C
′) > w(B). But then w(C′) > w(C), a
contradiction of the first lemma. �
Corollary 9.3.8 Let (s, t) > (s1, t1) be elements of S
′, and B, B1 be blocks of
S such that (s, t) ∈ B′, and (s1, t1) ∈ B
1. Then w(B) > w(B1).
Proof: Let w(B) = (r, c) and w(B1) = (r1, c1). By Corollary 9.3.6, we have
four possibilities. Since (r, c) dominates (s, t) and (r1, c1) dominates (s1, t1),
the possibilities c < r < c1 < r1 and c1 < r1 < c < r are eliminated. It is
thus enough to eliminate the possibility c1 < c < r < r1. Suppose that this
is the case. Then, by Corollary 9.3.7, j1 < j, where j1 and j are such that
B ⊆ Sj and B1 ⊆ Sj1 . Now, by Lemma 9.3.5 (2), there exists α in Sj1 such
that α > (s, t) > (s1, t1). But then this contradicts Lemma 9.3.2. �
Corollary 9.3.9 For a B ⊆ Si of S, the depth of w(B) in Sw is exactly i.
Proof: That the depth is at least i follows from Lemma 9.3.5. That the depth
cannot exceed i follows from Corollary 9.3.7. �
Corollary 9.3.10 Let α ∈ S′k(up), β ∈ S
m(up), and α > β. Then k < m.
Proof: Corollary 9.3.8 and Corollary 9.3.9. �
9.4 More lemmas
This subsection is a collection of lemmas to be invoked in the later subsec-
tions. More specifically, Lemma 9.4.1 and Corollary 9.4.2 are invoked in the
proof of Proposition 4.1.1 in §10.1, Lemma 9.4.3 in the proof of the first half of
Proposition 4.1.2 in §10.2, and Lemma 9.4.4 in the proof of the second half of
Proposition 4.1.2 in §10.3. Throughout this subsection, S denotes a monomial
in ON.
Lemma 9.4.1 Let C be a v-chain in S′, α an element of C, and α′ ∈ SC,α.
Then depth
α′ ≤ k(even), for k the integer such that α ∈ S′k(up).
Proof: Proceed by induction on k. If k = 1, the assertion follows from Corol-
lary 9.3.10, so assume k > 1. Choose a v-chain C′ in SC with tail α
′ and
depthC′α
′ = depth
(α′). The length of a v-chain in SC,α is clearly at most 2.
So, if γ′ is the element two steps before α′ in C′ (if γ′ does not exist then there is
clearly nothing to prove), then γ′ ∈ SC,γ with γ > α (see Proposition 5.3.4 (2)).
We claim that depth
(γ′) ≤ k(odd) − 1. It is enough to prove the claim, for
then depth
(α′) = depthC′α
′ = depthC′γ
′ + 2 ≤ k(odd)− 1 + 2 = k(even).
The claim follows by induction from Corollary 9.3.10 if k is odd or more
generally if γ ∈ S′l(up) with l ≤ k(odd) − 1. So assume that k is even and
γ ∈ S′k−1(up). By 7.5.4, it is not possible that γ is of type H and ph(γ) > α.
So the only possibility is that α′ = ph(α) and γ > α is connected. In particular,
γ is of type V and α of type H in C and γ′ = pv(γ).
Now let µ be the first element in the connected component of α in C. The
cardinality of the part µ > . . . > γ of C is even (by Proposition 5.3.1 (1), it
follows that the cardinality of µ > . . . > α is odd), say e. Letting m be such
that µ ∈ S′m(up), we have, by Proposition 9.3.10, m ≤ k − 1 − (e − 1) =
k − e. If m(even) < k − e, then, since depthC′γ
′ = depthC′pv(µ) + e − 1 (by
Proposition 5.3.4 (1), since, by Proposition 5.3.1 (2), µ, . . . , γ all have type V
in C) and depthC′pv(µ) ≤ m(even) by induction, it follows that depthC′γ
k − e+ e− 1 = k − 1, and we are done.
So suppose that m(even) = k − e. Let ν be the element just before µ in C
(if such an element does not exist, then depthC′γ
′ = e ≤ k − 2—observe that
m(even) ≥ 2—and we are done). Then ν > µ is not connected (by choice of µ).
So ph(ν) > µ. By Proposition 7.5.4, this means that j ≤ m(even) − 2 where j
is the odd integer defined by ν ∈ S′j(up) ∪ S
j+1(up). So, again by induction,
depthC′γ
′ = depthC′ph(ν) + e ≤ m(even) − 2 + e = k − 2, and the claim is
proved. �
Corollary 9.4.2 The O-depth of an element α in S′ is at most k where k is
such that α ∈ S′k(up).
Proof: Let C′ be a v-chain in SC with tail qC,α. If k is even, then, by the
lemma, depthC′qC,α ≤ k. So suppose that k is odd. Let γ
′ be the immediate pre-
decessor of qC,α in C
′. By Proposition 5.3.4 (2), γ > α, and so γ ∈ S′l(up) with
l ≤ k− 1 (see the observation in the first paragraph of the proof of the lemma).
So depthC′γ
′ ≤ k − 1 (by the lemma) and depthC′α
′ = depthC′γ
′ + 1 ≤ k. �
Lemma 9.4.3 Let S be a monomial in ON and Oπ(S) = (w,S′). Let i <
k be integers, α an element of S′i(up), and δ an element of (Sw)k(up) that
dominates α.
1. If k is even, then there exists β ∈ S′k(up) with α > β.
2. If k is odd and wk,k+1 O-dominates the singleton v-chain α, then either
there exists β ∈ S′k(up) with α > β or there exists γ ∈ S
k+1(up) with
ph(α) > γ.
Proof: Write α = (r, c) and δ = (A,B). By Corollary 9.3.9, there exists
a block B of Sk such that δ = w(B). Let (D,B) be the first element of B
(arranged in increasing order of row and column indices). We have the following
possibilities:
(i) D ≤ A and (D,B) ∈ S
(ii) k is odd, S is truly orthogonal at k, (D,B) = (A,B) = pv(σk), and B
consists of the single diagonal element (D,B) = (B∗, B).
(iii) k is even, S is truly orthogonal at k−1, (D,B) = (A,B) = ph(σk−1), and
B consists of the single diagonal element (D,B) = (B∗, B).
We claim the following: in case (i), D < r (in particular, D < A); in case (ii),
the row index of σk is less than r; and case (iii) is not possible. The first
two assertions and also the third in the case i < k − 1 follow readily from
Lemma 9.3.3; in case (iii) holds and i = k − 1, then σk−1 > α, a contradiction
to Lemma 9.3.2.
First suppose that possibility (ii) holds. Write σk = (s,B). Since s < r
and ph(σk) ∈ N, it is clear that ph(α) = (r, r
∗) also belongs to N. From the
hypothesis that wk,k+1 O-dominates {α}, it follows that there is an element
of (Sw)k+1 that dominates ph(α) = (r, r
∗). Such an element must be diagonal
(because of the distinguishedness ofSw), and so must be the w(C) for the unique
diagonal block C ofSk+1. In particular, this means that there are elements other
than (s, s∗) in Sk+1, and so S
k+1 is non-empty. In the arrangement of elements
of S′k+1(up) in increasing order of row and column numbers, let γ = (e, s
∗) be
the last element. Then e < s < r and r∗ < s∗, so ph(α) > γ, and we are done.
Now suppose that possibility (i) holds. Let (p, q) be the element of Sk such
that p is the largest row index that is less than r, and, among those elements
with row index p, the maximum possible column index is q. The arrangement of
elements of Sk (in increasing order or row and column indices) looks like this:
. . . , (p, q), (s, t), . . .
Since p < r ≤ A and w(B) = (A,B), we can be sure that (p, q) is not the last
element of B.
We first consider the case c < t. Then α = (r, c) > β := (p, t) ∈ S′k.
If β ∈ S′k(up), then we are done. It is possible that (p, q) lies on or below
the diagonal so that β lies below the diagonal, in which case, α > β(up) and
β(up) ∈ S′k(up), and again we are done.
Now suppose that t ≤ c. We claim that:
• (s, t) belongs to the diagonal;
• k is odd and S is truly orthogonal at k; and
• σk = (u, t) with u < r.
Suppose that (s, t) does not belong to the diagonal. Since r ≤ s (by choice of
(p, q)), it follows that (s, t) dominates (r, c). This leads to a contradiction to
Lemma 9.3.3, for either (s, t) or its reflection (t∗, s∗) (whichever is above the
diagonal) belongs to S
and dominates α = (r, c) in S′i(up). This shows that
(s, t) belongs to the diagonal. If k is even, then (s, t) = ph(σk−1), which means
σk−1 > α, again contradicting Lemma 9.3.3, so k must be odd. It also follows
that S is truly orthogonal at k and that (s, t) = pv(σk). Writing σk = (u, t), if
r ≤ u, then σk would dominate α, again contradicting Lemma 9.3.3. So u < r,
and the claim is proved.
To finish the proof of the lemma, now proceed as in the proof when possi-
bility (ii) holds. �
Lemma 9.4.4 Let T be a monomial in ON and w an element of I(d) that O-
dominates T. Let β′ > β be elements Sw(up). Let d−1 and d be their respective
depths in Sw. Let α be an element of OP
β or more generally an element of
ON such that
(a) it is dominated by β,
(b) it is not comparable to any element of Pβ, and
(c) in case d is odd, then {α} ∪ Tw,d,d+1 has O-depth at most 2.
1. there exists α′ ∈ P∗β′(up) with α
′ > α;
2. for α′ as in (1), if α′ is diagonal, then ph(δd−2) > α if d is odd and
δd−1 > α if d is even.
Proof: Assertion (2) is rather easy to prove. If d is odd, then, in fact,
ph(δd−2) = α
′; if d is even, then δd−1 has the same column index as α
′ and, by
Proposition 8.2.2 (4), has row index more than that of α, so δd−1 > α.
Let us prove (1). Write α = (r, c), β = (R,C), and β′ = (R′, C′). There
exists, by the definition of P∗β′ , an element in P
β′ with column index C
′. We
have C′ < c (for C′ < C ≤ c). Let (r′, c′) be the element of P∗β′ such that c
is maximum possible subject to c′ < c and among those elements with column
index c′ the maximum possible row index is r′. If r < r′, then we are done (if
(r′, c′) is below the diagonal, its mirror image would have the desired properties).
It suffices therefore to suppose that r′ ≤ r and arrive at a contradiction.
In the arrangement of elements of P∗β′ in non-decreasing order of row and
column indices, there is a portion that looks like this:
. . . , (r′, c′), (a, b), . . .
Since there is in P∗β an element with row index R
′ (and clearly r′ ≤ r < R < R′),
it follows that (a, b) exists (that is, (r′, c′) is not the last element in the above
arrangement). It follows from the construction of P∗β′ from Pβ′ that (r
′, b) is an
element in Pβ′ . By the choice of (r
′, c′), we have c ≤ b. Thus (r, c) dominates
(r′, b).
The proof now splits into two cases accordingly as d is even or odd. First
suppose that d is even. Then, since β dominates (r′, b) and yet (r′, b) does not
belong to Pβ, there exists a v-chain in Tw,d−1,d of length 2 and head (r
′, b). The
tail of this v-chain then belongs toPβ and is dominated by (r, c), a contradiction
to our assumption that α is not comparable to any element of Pβ.
Now suppose that d is odd. Choose a v-chain C in T with head (r′, b) that
is not O-dominated by wd. Let D be the part of C consisting of elements of
O-depth (in C) at most 2. We claim that D is O-dominated by wd,d+1. In fact,
we claim the following: Any v-chain F with head (r′, b) and O-depth at most 2
is O-dominated by wd,d+1.
To prove the claim, we first prove the following subclaim:
(†) If the horizontal projection of (r′, b) belongs to N, then β is on
the diagonal and dominates the vertical projection of (r′, b), and the
diagonal element β1 of (Sw)d+1 dominates the horizontal projection
of (r′, b).
Let ph(r
′, b) ∈ N. Then β belongs to the diagonal because Sw is distinguished
and symmetric. Once β is on the diagonal, it is clear that it dominates pv(r
′, b)
(from our assumptions, β dominates (r, c) and (r, c) dominates (r′, b)). It follows
from Proposition 8.2.2 (3) that the row index of β1 exceeds the row index r of
(r, c), so β1 dominates ph(r
′, b). This finishes the proof of the subclaim (†).
To begin the proof of the claim, observe that F has length at most 2. Suppose
first that F consists only of the single element (r′, b). The type of (r′, b) in
F is either H or S. If it is S, then since β dominates (r′, b), the claim follows
immediately. If it is H, then the claim follows immediately from the subclaim (†).
Continuing with the proof of the claim, let now F consist of two elements:
(r′, b) > µ. Let γ be the element of Sw such that µ ∈ Pγ , and let e be the
depth of γ in Sw. From Lemma 8.2.1 (2b) it follows that e ≥ d. If e = d, then
γ = β (by the distinguishedness of Sw), and the comparability of (r, c) and µ
contradicts our hypothesis (b). So e ≥ d+1, and there exists δ of depth d+1 in
Sw that dominates µ. We have β > δ (again by the distinguishedness of Sw).
The possibilities for the types of (r′, b) and µ in F are: S and S, V and V, H
and S (in the last case ph(r
′, b) 6> µ by Lemma 6.3.1 (1)). Noting the existence
in (Sw)d,d+1 of the v-chain β > δ in the first case and also of β > β1 (where
β1 is as in the subclaim) in the last case, the proof of the claim in these cases
is over. So suppose that the second possibility holds. The distinguishedness of
Sw implies that δ = β1. Since δ is diagonal, it dominates the vertical projection
of µ. Noting the existence of the v-chain in β > δ in (Sw)d,d+1, the proof of
the claim in this case too is over.
We continue with the proof of the lemma. It follows from the claim that D is
O-dominated by wd,d+1. From Corollary 9.2.3 it follows that the complement E
of D in C is not O-dominated by wd+2,d+3 (in particular, that E is non-empty)
and that every v-chain in T with head ǫ (where ǫ denotes the head of E) is
O-dominated by wd (given such a v-chain, the concatenation of D with it is
O-dominated by wd−2, and ǫ continues to have O-depth 3 in the concatenated
v-chain). Thus ǫ belongs to Tw,d,d+1. From (1) and (2b) of Lemma 8.2.1 it
follows that the element µ of C in between (r′, b) and ǫ (if it exists at all) also
belongs to Tw,d,d+1. Now consider the v-chain obtained as follows: take the part
of C up to (and including) ǫ and replace its head (r′, b) by (r, c). This chain has
O-depth 3 and lives in {α} ∪ Tw,d,d+1, a contradiction to hypothesis (c). �
Corollary 9.4.5 Let T be a monomial in ON and w an element of I(d) that
O-dominates T. Let β′ > β be elements of Sw(up), α an element of OP
β, and
d′ := depth
1. If d′ is odd, there exists α′ ∈ OP∗β′ such that α
′ > α.
2. If there does not exist α′ ∈ OP∗β′ such that α
′ > α then (d′ is even by (1)
above and) there exists α′′ ∈ OP∗β′′ such that ph(α
′′) > α, where β′′ is the
unique element of (Sw)d′−1 such that β
′′ > β′.
Proof: Immediate from the lemma. �
Corollary 9.4.6 Let T be a monomial in ON and w an element of I(d) that
O-dominates T. Let β, β′ be elements of Sw(up), and α, α
′ elements of OP∗β
and OP∗β′ respectively.
1. If α′ > α then β′ > β (in particular, depth
β′ < depth
2. If ph(α
′) > α and depth
β is even, depth
β′ ≤ depth
β − 2.
Proof: (1) Writing β = (r, c) and β′ = (r′, c′), there are, since both β and β′
dominate α and Sw is distinguished, the following four possibilities:
c < r < c1 < r1, c1 < r1 < c < r, c < c1 < r1 < r, c1 < c < r < r1
Since α′ > α, and α, α′ are dominated respectively by β, β′ (this is because α,
α′ belong to OP∗β, OP
β′ respectively), the possibilities c < r < c1 < r1 and
c1 < r1 < c < r are eliminated (by the distinguishedness of Sw). It is thus
enough to eliminate the possibility β > β′. Suppose, by way of contradiction,
that β > β′. By Corollary 9.4.5, either there exists γ ∈ OP∗β such that γ > α
in which case the v-chain γ > α in OP∗β contradicts Proposition 8.2.2 (3) or (4),
or d := depth
β is even and there exists (with β′′ being the unique element
in Sw such that β
′′ > β and depth
β′′ = d− 1) an element α′′ ∈ OP∗β′′ with
′′) > α′, in which case the v-chain α′′ > α in T⋆w,d−1,d has O-depth 3 and
so contradicts Proposition 8.2.2 (2).
(2) Set d := depth
β. If depth
β′ were d− 1, then the v-chain α′ > α in
T⋆w,d−1,d would be ofO-depth 3 and so would contradict Proposition 8.2.2 (2). �
10 The Proof
The aim of this section is to prove Propositions 4.1.1 and 4.1.2. The proof of
first proposition appears in §10.1 and that of the second in §§10.2, 10.3. In §9.4
some lemmas are established that are used in the proofs. Needless to say that
the lemmas maybe unintelligible until one tries to read the proofs in the later
subsections.
10.1 Proof of Proposition 4.1.1
(1) By definition, w is the element of I(d) associated to the distinguished mono-
mial ∪kSw(k). By the very definition of this association, we have w ≥ v.
(2) This follows from the corresponding property of the map π of [7]. More
precisely, that property justifies the third equality below. The other equalities
are clear from the definitions.
v-degree(w) + degree(S′) =
degree(Sw) +
degree(S′k)
degree(Sw(k)) + degree(S
degree(Sk)
j odd
degree(Sj,j+1)
j odd
degree(S
j ) + degree(S
= degree(S)
(3) We have:
w O-dominates S′ ⇔ w ≥ wC ∀ v-chain C in S
⇔ w dominates SC ∀ v-chain C in S
⇔ ∀ v-chain C in S′, ∀ α′ = (r, c) ∈ SC ,
∃ β = (R,C) ∈ Sw with C ≤ c, r ≤ R,
and depth
β ≥ depth
The first equivalence above follows from the definition of O-domination, the
second from [7, Lemma 4.5], the third from Lemma 9.1.4.
Now let C be a v-chain in S′ and α′ = (r, c) in SC . We will show that
there exists β in Sw that dominates α and satisfies depthSwβ ≥ depthSCα
Let α be the element in C such that α′ ∈ SC,α, let k be such that α ∈ S
k(up),
and let B be the block of Sk such that α ∈ B
′. Writing α = (r1, c1) and
w(B) = (R1, C1), we have C1 ≤ c1 and r1 ≤ R1 straight from the definition of
w(B). By Corollary 9.3.9, depth
w(B) = k.
First suppose that w(B) dominates α′ (meaning C1 ≤ c and r ≤ R1).
If k ≥ depth
α′, we are clearly done; by Corollary 9.4.2, this is the case
when α′ = qC,α. So suppose that α is of type H, α
′ = ph(α), and that k <
depth
α′. By Lemma 9.4.1, depth
α′ ≤ k(even). It follows that k is odd
and depth
α′ = k + 1. By Corollary 7.5.3, S is truly orthogonal at k, which
means that Sk+1 has a diagonal block, say C. Note that w(C) dominates ph(σk)
which in turn dominates ph(α). Since depthSww(C) = k+1 by Corollary 9.3.9,
we are done.
Now suppose that w(B) does not dominate α′. Then B is non-diagonal and
α′ = pv(α). Since B is non-diagonal, ph(α) 6∈ N, and α cannot be of type H.
So α is of type V in C. It follows easily (see Proposition 5.3.1 (3)) that α is
the critical element in C, and and that last element in its connected component
in C; by Lemma 6.3.1 (4), O-depthC(α) = depthSC qC,α =: d is even. By
Proposition 5.3.1 (1), (2), the cardinality of the connected component of α in
C is even.
The immediate predecessor γ of α in C is connected to α (this follows from
what has been said above). It is of type V in C, ph(γ) belongs to N, and
depth
pv(γ) = d − 1 (see Lemma 6.3.1 (1)). Let ℓ be such that γ ∈ S
ℓ(up).
Let C be the block of Sℓ such that γ ∈ C
′. Since ph(γ) ∈ N, C is diagonal.
Note that w(C) dominates pv(γ) and that pv(γ) > pv(α). By Corollary 9.3.9,
depth
w(C) = ℓ. Thus if d ≤ ℓ we are done. On the other hand, d− 1 ≤ ℓ by
Corollary 9.4.2.
So we may assume that ℓ = d− 1. By Corollary 7.5.3, S is truly orthogonal
at d − 1. This implies that Sd has a diagonal block, say D. Note that w(D)
dominates ph(σd−1) which in turn dominates ph(γ). Writing γ = (r2, c2), since
γ > α is connected, it follows that (r1, r
2) belongs to ON. Now both w(B) and
w(D) dominate (r1, r
2). Since Sw is distinguished and symmetric and w(B) is
not on the diagonal, it follows that w(D) > w(B). This implies, since w(D) is
on the diagonal, w(D) > pv(α). Since depthSww(D) = d by Corollary 9.3.9, we
are done.
(4) Let x be an element of I(d) that O-dominates S. We will show that
x ≥ w. By [7, Lemma 5.5], it is enough to show that x dominates Sw. By
Lemma 9.1.4, it is enough to show the following: for every block B of S, there
exists β in Sx such that β dominates w(B) and depthSxβ ≥ depthSww(B).
Let B be a block of S. By Corollary 9.3.9, depth
w(B) = k where B ⊆
Sk. Let S
x denote the set of elements of Sx of depth at least k. Our goal is
to show that there exists β in Skx that dominates w(B). It follows easily from
the distinguishedness of Sx and the fact that B is a block, that it suffices to
show the following: given α ∈ B, there exists β in Skx (depending upon α) that
dominates α. Moreover, since B and Skx are symmetric, we may assume that
α = α(up).
So now let α = α(up) belong to B. Then either
1. α belongs to S
2. k is odd, S is truly orthogonal at k, and α = pv(σk), or
3. k is even, S is truly orthogonal at k − 1, and α = ph(σk−1).
The proofs in the three cases are similar. In the first case, choose a v-chain C
in S with tail α such that O-depthC(α) = k (see Corollary 6.1.3 (1)). Then
depth
qC,α = k and, clearly, qC,α dominates α. Since x dominates SC , there
exists, by Lemma 4.5 of [7], β in Skx that dominates qC,α (and so also α).
In the second case, choose a v-chain C in S with tail σk with the property
that O-depthC(σk) = k. Then depthSC qC,σk = k. Since ph(σk) belongs to N,
σk is of type V or H in C, so qC,σk = α. Since x dominates SC , there exists, by
[7, Lemma 4.5], β in Skx that dominates qC,σk = α.
In the third case, choose a v-chain C in S with tail σk−1 such that the
O-depth in C of σk−1 is k − 1. Then depthSCqC,σk−1 = k − 1. Since ph(σk−1)
belongs to N, σk−1 is of type V or H in C, so qC,σk−1 = pv(σk−1). From
Lemma 6.3.1 (4), it follows, since k − 1 is odd, that σk−1 is of type H. Since
pv(σk−1) > ph(σk−1) = α, it follows that depthSCph(σk−1) ≥ k (in fact equality
holds as is easily seen). Since x dominates SC , there exists, by [7, Lemma 4.5],
β in Skx that dominates ph(σk−1) = α. �
10.2 Proof that OφOπ = identity
LetS be a monomial inON and let Oπ = (w,S′). We need to show thatOφ ap-
plied to the pair (w,S′) gets us back toS. We know from (3) of Proposition 4.1.1
that w O-dominates S′, so Oφ can indeed be applied to the pair (w,S′).
The main ingredients of the proof are the corresponding assertion in the
case of Grassmannian [7, Proposition 4.2] and the following claim which we will
presently prove:
(S′)w,j,j+1 = S
j(up) ∪S
j+1(up) for every odd integer j
Let us first see how the assertion follows assuming the truth of the claim, by
tracing the steps involved in applying Oφ to (w,S′). From the claim it follows
that when we partition S′ into pieces (see §8), we get S′j(up) ∪S
j+1(up) (for
odd integers j). Adding the mirror images will get us to S′j ∪ S
j+1. From
Corollary 9.3.9 it follows that wj,j+1 is exactly the element of I(d, 2d) obtained
by acting π on Sj ∪ Sj+1. Now, since φ ◦ π = identity, it follows that on
application of φ to (wj,j+1,S
j+1) we obtain Sj ∪Sj+1. By twisting the
two diagonal elements in Sj ∪ Sj+1 (if they exist at all) and removing the
elements below the diagonal d, we get back S
j,j+1. Taking the union of S
j,j+1
(over odd integers j), we get back S.
Thus we need only prove the claim. Since S′ is the union over all odd
integers of the right hand sides (this follows from the definition of S′), and the
left hand sides as j varies are mutually disjoint, it is enough to show that the
right hand side is contained in the left hand side. Thus we need only prove: for
j an odd integer and α an element in S′j(up) ∪S
j+1(up),
• every v-chain in S′ with head α is O-dominated by wj .
• there exists a v-chain in S′ with head α that is not O-dominated by wj+2.
To prove the first item, write T = Sj,j+1 := {α ∈ S|O-depth
(α) ≥ j}
and set Oπ(T) = (x,T′). By Proposition 6.3.6, we have T
i ∪ T
i+1 = S
i+j−1 ∪
i+j for any odd integer i. Thus, by the description of Oπ, we have T
∪k≥jS
k(up). By Corollary 9.3.9 and the description of Oπ, we have x = w
By Corollary 9.3.10, any v-chain inS′ with head belonging toS′j(up)∪S
j+1(up)
is contained entirely in ∪k≥jS
k(up). Finally, by Proposition 4.1.1 (3) applied
to T, the desired conclusion follows.
To prove the second item we use Lemma 9.4.3. Proceed by decreasing induc-
tion on j. For j sufficiently large the assertion is vacuous, for S′j(up)∪S
j+1(up)
is empty. To prove the induction step, assume that the assertion holds for j+2.
If the v-chain consisting of the single element α is not O-dominated by wj+2,
then we are done. So let us assume the contrary. Since the O-depth of the
singleton v-chain α is at most 2, it follows from Lemma 9.2.2 that wj+2,j+3 O-
dominates the v-chain α. Apply Lemma 9.4.3 with k = j+2. By its conclusion,
either there exists β ∈ S′j+2(up) such that α > β or there exists γ ∈ S
j+3(up)
such that ph(α) > γ.
First suppose that a γ as above exists. By induction, there exists a v-chain
in S′—call it D—with head γ that is not O-dominated by wj+4. Let C be the
concatenation of α > γ and D. Since elements of D haveO-depth at least 3 in C
(Lemma 6.3.1 (1)), it follows from Corollary 9.2.3 that C is not O-dominated
by wj+2, and we are done.
Now suppose that such a γ does not exist. Then a β as above exists. If
α > β is not O-dominated by wj+2 we are again done. So assume the contrary.
Since the O-depth of β in α > β is at least 2, it follows that there exists an
element of (Sw)j+3 that dominates β. Applying Lemma 9.4.3 again, this time
with k = j + 3, we find γ′ ∈ S′j+3(up) such that β > γ
′. Arguing as in the
previous paragraph with γ′ in place of γ, we are done. �
10.3 Proof that OπOφ = identity
Let T be a monomial in ON and w an element of I(d) that O-dominates T. We
can apply Oφ to the pair (w,T) to obtain a monomial T⋆w in ON. We need to
show that Oπ applied to T⋆w results in (w,T). The main step of the proof is to
establish the following:
T⋆w,j,j+1 = (T
j,j+1 (10.3.1)
(for the meaning of the left and right sides of the above equation, see §8 and
§7 respectively). Assuming this for the moment let us show that Oπ ◦ Oφ =
identity.
We trace the steps involved in applying Oπ to T⋆w. From Eq. (10.3.1) it
follows that when we break up T⋆w according to the O-depths of its elements
as in §7, we get T⋆w,j,j+1 (as j varies over odd integers). The next step in the
application of Oπ is the passage from (T⋆w)
j,j+1 to (T
w)j,j+1. This involves
replacing σj by its projections and adding the mirror image of the remaining
elements of (T⋆w)
j,j+1. It follows from Proposition 8.2.2 (3) that σj = δj and so
(T⋆w)j,j+1 = (Tw,j,j+1 ∪ T
w,j,j+1)
⋆. The next step is to apply π to (Tw,j,j+1 ∪
w,j,j+1)
⋆. Since π is the inverse of φ (as proved in [7]), we have π((Tw,j,j+1 ∪
w,j,j+1)
⋆) = (wj,j+1,Tw,j,j+1). Since Sw and T are respectively the unions, as
j varies over odd integers, of (Sw)j,j+1 and Tw,j,j+1, we see that Oφ applied to
T⋆w results in (w,T).
Thus it remains only to establish Eq. (10.3.1). It is enough to show that the
left hand side is contained in the right hand side, for the union over all odd j of
either side is T⋆w and the right hand side is moreover a disjoint union. In other
words, we need only show that the O-depth in T⋆w of an element of T
w,j,j+1
is either j or j + 1. We will show, more precisely, that, for any element β of
Sw, the O-depth in T
w of any element of OP
β equals the depth in Sw of β.
Lemma 9.4.4 will be used for this purpose.
Let α be an element of OP∗β and set e := O-depthT⋆w (α). We first show,
by induction on d := depth
β, that e ≥ d. There is nothing to prove in case
d = 1, so we proceed to the induction step. Let β′ be the element of Sw of depth
d − 1 such that β′ > β. If there exists α′ in OP⋆β′ with α
′ > α, the desired
conclusion follows from Corollary 6.1.3 (3) and induction. Lemma 9.4.4 says
that such an α′ exists in case d is even. So suppose that d is odd and such an
α′ does not exist. The same lemma now says that ph(δd−2) > α, so the desired
conclusion follows from Lemma 6.3.1 (1).
We now show, by induction on e, that d ≥ e. There is nothing to prove
in case e = 1, so we proceed to the induction step. Let C be a v-chain in T⋆w
with tail α and having the good property of Proposition 6.3.3. Let α′ be the
immediate predecessor in C of α. Let β′ in Sw be such that α
′ ∈ OP∗β′ (we are
not claiming at the moment that β′ is unique although that is true and follows
from the assertion that we are proving, the distinguishedness of Sw, and the
fact that β′ dominates α′). It follows from Corollary 9.4.6 that β′ > β.
Let d′ := depth
β′. It follows from Corollary 6.1.3 (3) that e′ < e where
e′ := O-depth
(α′). We have, d ≥ d′+1 ≥ e′+1 ≥ (e−2)+1 = e−1, the first
equality being justified because β′ > β, the second by the induction hypothesis,
and the last by Lemma 6.3.1 (1). It suffices to rule out the possibility that
d = e − 1. So assume d = e − 1. Then d = d′ + 1 and d′ = e′ = e − 2. It
follows from (1) of Lemma 6.3.1 that the v-chain α′ > α has O-depth 3 and
from (3) of the same lemma that e′ is odd. But then we get a contradiction to
Proposition 8.2.2 (2) (α′ and α belong to T⋆w,d′,d′+1). The proof of Eq. (10.3.1)
is thus over. �
10.4 Proof of Proposition 4.1.3
Observe that the condition (‡) makes sense also for a monomial of N. By virtue
of belonging to I(d), v has f∗ as an entry. It follows from the description of
the bijection w ↔ Sw of §5.1.2 that for an element w of I(d) to satisfy (‡) it is
necessary and sufficient that Sw (equivalently all its parts Sw,j,j+1) satisfy (‡).
(1) Since T satisfies (‡), so do its parts Tw,j,j+1 and Tw,j,j+1 ∪ T
w,j,j+1
(adding the mirror image preserves (‡)). Since Sw,j,j+1 also satisfies (‡), it
follows from the description of the map φ of [7] (observe the passage from a
piece P to its “star” P∗) that the (Tw,j,j+1 ∪ T
w,j,j+1)
⋆ satisfy (‡). Since
the “twisting” involved in the passage from (Tw,j,j+1 ∪ T
w,j,j+1)
⋆ to T⋆w,j,j+1
involves only a rearrangement of row and column indices, it follows that the
T⋆w,j,j+1 satisfy (‡). Finally so also does their union T
(2) The parts S
j,j+1 of S clearly satisfy (‡). Therefore so do the Sj,j+1, for,
first of all, adding the mirror image preserves (‡), and then the removal of σj
and addition of its projections involves only a rearrangement of row and column
indices. It follows from description of the map π of [7] (observe the passage from
a block B to the pair (w(B),B′)) that both Sw,j,j+1 and S
j,j+1 satisfy (‡).
Finally, Sw and S
′ being the union (respectively) of Sw,j,j+1 and S
j,j+1(up),
they satisfy (‡). �
Part IV
An Application
As an application of the main theorem (Theorem 2.3.1), an interpretation of
the multiplicity is presented.
11 Multiplicity counts certain paths
Fix elements v, w in I(d) with v ≤ w. It follows from Corollary 2.3.2 that the
multiplicity of the Schubert variety X(w) in Md(V ) at the point e
v can be in-
terpreted as the cardinality of a certain set of non-intersecting lattice paths. We
first illustrate this by means of two examples and then justify the interpreta-
tion.
11.1 Description and illustration
The points of N can be represented, in a natural way, as the lattice points of
a grid. The column indices of the points of the grid are the entries of v and
the row indices are the entries of {1, . . . , 2d} \ v. In Figure 11.1.1 the points
of ON and those of the diagonal in N are shown (for the specific choice of v in
Example 11.1.1). The open circles represent the points of Sw(up), where Sw
is the distinguished monomial in N that is associated to w as in §5.1.2. From
each point β of Sw(up) we draw a vertical line upwards from β and let β(start)
denote the top most point of ON on this line. In case β is not on the diagonal,
draw also a horizontal line rightwards from β and let β(finish) denote the right
most point of ON on this line. In case β is on the diagonal, then β(finish) is
not a fixed point but varies subject to the following constraints:
• β(finish) is one step away from the diagonal (that is, it is of the form (r, c),
for some entry c of v, where r is the largest integer less than c∗ that is not
an entry of v);
• the column index of β(finish) is not less than that of β;
• if depth
β is odd, then the horizontal projection of β(finish) is the same
as the vertical projection of γ(finish) where γ is the diagonal element of
Sw of depth 1 more than that of β.
With v and w as in Example 11.1.1, we have β(start) = (6, 3) and β(finish) =
(9, 5) for β = (9, 3); β(start) = β(finish) = (21, 20) for β = (21, 20); β(start) =
(15, 11) for the diagonal element β = (36, 11); β(start) = (6, 1) for the diagonal
element β = (46, 1). In the particular case (of non-intersecting lattice paths)
drawn in Figure 11.1.1, β(finish) = (27, 19) for β = (36, 11) and β(finish) =
(28, 14) for β = (46, 1).
A lattice path between a pair of such points β(start) and β(finish) is a se-
quence α1, . . . , αq of elements of ON with α1 = β(start) and αq = β(finish)
such that, for 1 ≤ j ≤ q − 1, if we let αj = (r, c), then αj+1 is either (R, c) or
(r, C) where R is the least element of {1, . . . , 2d} \ v that is bigger than r and
C the least element of v that is bigger than c. Note that if β(start) = (r, c) and
β(finish) = (R,C), then q equals
|({1, . . . , 2d} \ v) ∩ {r, r + 1, . . . , R}|+ |v ∩ {c, c+ 1, . . . , C}| − 1,
where | · | is used to denote cardinality.
Consider the set Pathsw of all tuples (Λβ)β∈Sw(up) of paths where
• Λβ is a lattice path between β(start) and β(finish) (if β is on the diagonal,
then β(finish) is allowed to vary in the manner described above);
• Λβ and Λγ do not intersect for β 6= γ.
The number of such p-tuples, where p := |Sw(up)|, is the multiplicity of X(w)
at the point ev.
Example 11.1.1 Let d = 23,
v = (1, 2, 3, 4, 5, 11, 12, 13, 14, 19, 20, 22, 23, 26, 29, 30, 31, 32, 37, 38, 39, 40, 41),
w = (4, 5, 9, 10, 14, 17, 18, 21, 23, 25, 27, 28, 31, 32, 34, 35, 36, 39, 40, 41, 44, 45, 46),
so that
Sw ={(9, 3), (10, 2), (17, 13), (18, 12), (21, 20), (25, 22), (27, 26),
(28, 19), (34, 30), (35, 29), (36, 11), (44, 38), (45, 37), (46, 1)}
and Sw(up) =
{(9, 3), (10, 2), (17, 13), (18, 12), (21, 20), (25, 22), (28, 19), (36, 11), (46, 1)}.
A particular element of Pathsw is depicted in Figure 11.1.1. �
Example 11.1.2 Figure 11.1.2 shows all the elements of Pathsw in the follow-
ing simple case:
d = 7, v = (1, 2, 3, 4, 7, 9, 10), and w = (4, 6, 7, 10, 12, 13, 14).
We have Sw = {(6, 3), (12, 9), (13, 2), (14, 1)}, Sw(up) = {(6, 3), (13, 2)(14, 1)}.
There are 15 elements in Pathsw and thus the multiplicity in this case is 15. �
Example 11.1.3 Let d = 10,
v = (1, 2, 3, 4, 6, 8, 11, 12, 14, 16), and w = (8, 9, 11, 14, 15, 16, 17, 18, 19, 20).
so that Sw = {(20, 1)(19, 2)(18, 3), (17, 4), (9, 6)(15, 12)}. Figure 11.1.3 shows
a tuple of paths that is disallowed (meaning one that is not in Pathsw). The
elements of ON are represented as usual by a grid. The slanted line represents
1 2 3 4 5
11 12 13 14
19 20
22 23
Figure 11.1.1: An element of Pathsw with v and w as in Example 11.1.1
the diagonal d. The solid dot represents the point of Sw(up) that is not on d,
and the crosses on d represent the points of Sw(up) that lie on d. The tuple is
disallowed because the horizontal projection of the last point of the path Λβ1
is not the vertical projection of the last point of the path Λβ2 , where β1 =
(20, 1) and β2 = (19, 2) are the diagonal elements of Sw of depths 1 and 2
respectively. �
11.2 Justification for the interpretation
We now justify the interpretation in the previous subsection of the multiplicity.
Corollary 2.3.2 says that the multiplicity is the number of monomials in OR
of maximal cardinality that are square-free and O-dominated by w. Any such
monomial contains OR \ON, for, by the definition of O-domination, adding or
removing elements of OR\ON to or from a monomial does not alter the status
of its O-domination. One could therefore equally well consider the number of
monomials in ON of maximal cardinality that are square-free and O-dominated
✲ ✲ ✲ ✲ ✲
❄ ❄ ❄ ❄ ❄
❄ ❄ ❄ ❄ ❄
❄ ❄ ❄
Figure 11.1.2: All the 15 non-intersecting lattice paths of Example 11.1.2
diagonal
1        2        3        4        6        8        11      12      14      16     
The horizontal projection of the last point of the path associated to (20,1) is
not the same as the vertical projection of the last point of the path associated
to (19,2).    This is the reason that this tuple is disallowed.
Figure 11.1.3: A disallowed tuple of lattice paths (see Example 11.1.3)
by w. We now establish a bijection between the set Monw of such monomials
and the set Pathsw of non-intersecting lattice paths as in §11.1.
Each element Λ of Pathsw can be thought of, in the obvious way, as a
monomial in ON. We will continue to denote the corresponding monomial by Λ.
It is clear that the monomial Λ is square-free and that all such monomials Λ
have the same cardinality (in particular, that if Λ1 ⊆ Λ2 for two such monomials
then Λ1 = Λ2). In order to establish the bijection it therefore suffices to prove
the following proposition.
Proposition 11.2.1 1. w is the element of I(d) obtained on application of
Oπ to the monomial Λ (in particular (see Proposition 4.1.1), the monomial
Λ is O-dominated by w).
2. Given a monomial T of ON that is square-free and O-dominated by w,
there exists Λ such that T ⊆ Λ.
Proof: (1) Write Λ = (Λβ)β∈Sw(up). From the description of the map Oπ
in §7, it follows that it suffices to show that Λ
(in the notation of §7) is the
union ∪Λβ where β runs over all elements of depth k in Sw(up). In other words,
it suffices to show that the O-depth in Λ of any element of Λβ equals the depth
in Sw of β. To prove this, we observe the following (these assertions are easily
seen to be true thinking in terms of pictures): for fixed β ∈ Sw(up) and α ∈ Λβ ,
(A) For β′ in Sw(up) such that β
′ > β, there exists α′ ∈ Λβ′ such that α
′ > α.
(B) If α′ > α for some α′ in Λβ′ for some β
′ in Sw(up), then β
′ > β. If,
furthermore, β and β′ are diagonal, their depths in Sw are 1 apart, and
the depth in Sw of β is even, then the following is not possible: ph(α
belongs to N and ph(α
′) > α.
From (A) it is immediate that the O-depth e in Λ of an element α of Λβ is not
less than the depth d in Sw of β. We now show, by induction on e, that e ≤ d.
For e = 1 there is nothing to show. Suppose that e ≥ 2. Let C be a v-chain
in Λ having tail α and the good property of Proposition 6.3.3, α′ the immediate
predecessor in C of α, e′ the O-depth of α′ in Λ, β′ the element of Sw(up)
such that α′ ∈ Λβ′ , and d
′ the depth in Sw of β
′. From Corollary 6.1.3 (3) it
follows that e′ ≤ e − 1, so we may apply induction. From (B) it follows that
d′ ≤ d − 1, so that, by induction, e′ ≤ d − 1. If e′ ≤ d − 2, then we are done
by Lemma 6.3.1 (1). So suppose that e′ = d′ = d − 1. If d is odd, then the
conclusion e ≤ d follows from (1) and (3) of the same lemma. In case d is even,
then it follows from condition (B) and (1) of the same lemma.
(2) Let T be a square-free monomial in ON that is O-dominated by w. To
construct Λ such that T ⊆ Λ, we construct the “components” Λβ . As in §8, let
Pβ denote the piece of T corresponding to β ∈ Sw. From every point belonging
to Pβ(up) and also from β(start) carve out the South-West quadrant; if β is
not diagonal, then do this also from β(finish). The boundary of the carved out
portion (intersected with ON) gives a lattice path starting from β(start). In
case β is not diagonal, the path ends in β(finish). In this case as well as in
the case when β is diagonal and of even depth in Sw, we take Λβ to be this
lattice path. In case β is diagonal and of odd depth in Sw we do the carving
out from one more point before taking Λβ to be the boundary of the carved
out region, namely from the point that is one step away from the diagonal and
whose horizontal projection is the vertical projection of the end point of Λγ
where γ is the diagonal element of Sw of depth 1 more than β. We need to
justify why carving out from the extra point is still valid, and we do this now
by applying Lemma 8.1.4.
Let us first choose notation that is consistent with that of that lemma. Let
β and γ be diagonal elements in Sw of depths d and d + 1. Assume that d is
odd. Let the pieces of T corresponding to β and γ, when their elements are
arranged in increasing order of row and column indices, look like this:
. . . , (r1, a
∗), (a, r∗1), . . . ; . . . , (r2, b
∗), (b, r∗2), . . .
It is easy to see that the conditions on the numbers in the above display that pro-
vide the requisite justification are: r1 ≤ b and a
∗ < b∗ (if Pβ is empty then the
justification is easy). To prove that a∗ < b∗, observe that the diagonal elements
in P∗β and P
γ are respectively (a, a
∗) and (b, b∗), and apply Lemma 8.1.3 (2).
That r1 ≤ b now follows from Lemma 8.1.4 (1). This finishes the justification.
It suffices to prove the following claim: the lattice paths Λβ as β varies are
non-intersecting. Suppose that Λβ and Λβ′ intersect for β 6= β
′. Let α be a point
of intersection. Clearly β dominates all elements of Λβ and in particular α; for
the same reason β′ also dominates α. By the distinguishedness of Sw, we may
assume without loss of generality that β′ > β. It is easy to see graphically that
if γ in Sw is such that β
′ > γ > β then Λγ intersects either Λβ′ or Λβ : consider
the open portion of ON “caught between” the segment of Λβ′ from β
′(start) to
α and the segment of Λβ from β(start) to α; the starting point γ(start) of Λγ
lives in this region but its ending point does not (points strictly to the Northwest
of α can neither be of the form γ(finish) for γ not on the diagonal nor can they
be one step away from the diagonal); so Λγ must intersect one of the two lattice
path segments. We may therefore assume that the depths of β′ and β differ
by 1.
We now apply Lemma 9.4.4. From the construction of Λβ it readily fol-
lows that α satisfies the hypotheses (a), (b), and (c) of that lemma. By the
conclusion of Lemma 9.4.4, there exists α′ ∈ P∗β′(up) such that α
′ > α. On
the other hand, it follows from the construction of P∗β′ from Pβ′ , and from the
construction of Λβ′ that two elements one from P
β′ and another from Λβ′ are
not comparable. This is a contradiction to the comparability of α′ and α. �
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Index
>, relation on ON . . . . . . . . . . . . . . . . . 11
≤, partial order on I(d, 2d) . . . . . . . . 10
A, affine patch qv 6= 0 of Md(V ) . . . 14
α(down), for α ∈ N . . . . . . . . . . . . . . . . 25
α# for α in N . . . . . . . . . . . . . . . . . . . . . 22
α(up), for α ∈ N . . . . . . . . . . . . . . . . . . .25
anti-domination . . . . . . . . . . . . . . . . . . . 18
B, a specific Borel subgroup . . . . . . . . 8
β(finish), for β ∈ Sw(up) . . . . . . . . . . 65
β(start), for β ∈ Sw(up) . . . . . . . . . . . 65
block
in the sense of [7] . . . . . . . . . . . . .47
of a monomial S in ON . . . . . . . 51
comparability, of elements of R . . . . 29
connected components of a v-chain.23
connectedness of two succcessive ele-
ments in a v-chain . . . . . . . . 22
critical element (of a v-chain) . . . . . . 24
d, integral part of n/2 (unfortunately
also used otherwise) . . . . . 7, 9
degree, of a monomial . . . . . . . . . . . . . 10
degree, of a standard monomial . . . . 16
δj , for j odd . . . . . . . . . . . . . . . . . . . . . . . 44
depth (of an element α in a monomial
S in N) = depth
α . . . . . . 46
depth (of a monomial S in N) . . . . . 46
diagonal, d . . . . . . . . . . . . . . . . . . . . . 10, 11
distinguished (a subset of N) . . . . . . 21
domination (among elements of R) 29
domination map . . . . . . . . . . . . . . . . . . . 18
e1, . . . , en, a specific basis of V . . . . . . 8
ev, T -fixed point . . . . . . . . . . . . . . . . . . . 10
〈 , 〉, bilinear form on V . . . . . . . . . . . . 7
fθ := qθ/qv . . . . . . . . . . . . . . . . . . . . . . . . 14
head, of a v-chain. . . . . . . . . . . . . . . . . .11
horizontal projection ph(α) . . . . . . . . 22
I(d) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
I(d, 2d) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
i(even), for an integer i . . . . . . . . . . . . 29
i(odd), for an integer i . . . . . . . . . . . . . 29
intersection (of a monomial in a set
with a subset) . . . . . . . . . . . . 11
isotropic subspace . . . . . . . . . . . . . . . . . . 7
In . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
I ′n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
k, base field, (characteristic 6= 2) .7, 15
k∗(:= n+ 1− k) . . . . . . . . . . . . . . . . 7, 10
L, line bundle . . . . . . . . . . . . . . . . . . . . . 13
Λβ , for β ∈ Sw(up) . . . . . . . . . . . . . . . . 66
lattice path, from β(start) to β(finish),
denoted Λβ . . . . . . . . . . . . . . . 65
legs of α, for α ∈ ON . . . . . . . . . . . . . . 22
legs, intertwining of . . . . . . . . . . . . . . . .22
length, of a v-chain . . . . . . . . . . . . . . . . 11
Md(V ), orthogonal Grassmannian 7, 8
Md(V )
′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
monomial. . . . . . . . . . . . . . . . . . . . . . . . . .10
w . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
multiplicity, of X(w) at ev . . . . . . . . . 65
multiset := monomial . . . . . . . . . . . . . . 10
N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
n := dimV , (even from §2.1 on) . . 7, 9
O(V ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
O-depth . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
ON . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10
OP∗β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Oφ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17, 42
Oπ . . . . . . . . . . . . . . . . . . . . . . . . . 17, 34, 35
OR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10
O-domination . . . . . . . . . . . . . . . . . . . . . 11
orthogonal Grassmannian (Md(V )) . 7
Paths
w . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
Pβ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
P∗β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Pfaffian qθ . . . . . . . . . . . . . . . . . . . . . . . . . 14
ph(α), horizontal projection. . . . . . . .22
piece of T (see also caution) . . . . . . . 43
pθ, Plücker coordinate . . . . . . . . . . . . . 13
pv(α), vertical projection . . . . . . . . . . 22
qC,α, for α in a v-chain C . . . . . . . . . . 24
qθ, Pfaffian . . . . . . . . . . . . . . . . . . . . . . . . 14
R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
S, fixed monomial in ON in §7, §9.3 .
S, set of monomials in OR . . . . . . . . 18
modifications . . see Notation 4.2.1
SC , where C is a v-chain . . . . . . . . . . 23
SC,α, for α in a v-chain C . . . . . . . . . 24
Schubert varieties . . . . . . . . . . . . . . . . 7, 8
S(down), for a monomial S . . . . . . . 25
S#, for monomial S in N or R . . . . 22
σk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Sj,j+1, for S in ON, j odd . . . . . . . . 34
Sj,j+1(ext), for S in ON, j odd . . . 38
j,j+1, for S in ON, j odd . . . . . . . . 34
Sk, for monomial S in N . . . . . . . . . . 46
Sk, for monomial S in ON . . . . . . . . 34
Sk(ext), for monomial S in ON . . . 39
Sk,k+1, for monomial S in N . . . . . . 49
, for monomial S in ON . . . . . . . 34
SM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
modifications . . see Notation 4.2.1
SMv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
SMw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
SO(V ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
S′, for monomial S in ON . . . . . . . . 35
standard monomial . . . . . . . . . . . . . . . . 14
v-compatible . . . . . . . . . . . . . . . . . . 15
w-dominated . . . . . . . . . . . . . . . . . . 14
S(up), for a monomial S . . . . . . . . . . 25
Sj,j+1, for monomial S in ON . . . . 32
Sk, for monomial S in N . . . . . . . . . . 46
Sw(v)(m) . . . . . . . . . . . . . . . . . . . . . . . . . .11
Sw, w in I(d, 2d) or I(d, n) . . . . 21, 48
Sw,j,j+1, w in I(d), j odd . . . . . . . . . 42
Sjw, w in I(d), j odd . . . . . . . . . . . . . . 42
symmetric (monomial of N). . . . . . . .22
T , a specific maximal torus . . . . . . . . . 8
T , set of monomials in ON . . . . . . . . 18
modifications . . see Notation 4.2.1
tail, of a v-chain . . . . . . . . . . . . . . . . . . . 11
T, fixed monomial in ON in §8 . . . . . . .
truly orthogonal at j (j odd) . . . . . . 34
Tw,j,j+1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
(Tw,j,j+1 ∪ T
w,j,j+1)
⋆ . . . . . . . . . . . . . . 43
T⋆w,j,j+1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
T⋆w(:= Oφ(w,T)) . . . . . . . . . . . . . . . . . . 44
type (V, H, S), of an element in a v-
chain. . . . . . . . . . . . . . . . . .23–24
U , set of monomials in OR \ON . . 18
modifications . . see Notation 4.2.1
u∗, for u ∈ I(d) . . . . . . . . . . . . . . . . . . . . 19
V , vector space of dimension n . . . . . .7
v, fixed element of I(d) . . . . . . . . . . . . 10
v-chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
v-degree . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
vertical projection pv(α) . . . . . . . . . . . 22
w(C) (or wC), where C is v-chain. .11
w#, for w an element of I(d, 2d) . . . 21
w(k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34
(w,S′)(:= Oπ(S)), for S in ON . . 34,
w∗, for w an element of I(d, 2d) . . . .21
wj,j+1, j odd . . . . . . . . . . . . . . . . . . . . . . 42
wj , j odd . . . . . . . . . . . . . . . . . . . . . . . . . . 42
X(w), Schubert variety . . . . . . . . . . . . 10
xk, x
k, xk,k+1, for x ∈ I(d, n). . .48–49
Xr,c, variable . . . . . . . . . . . . . . . . . . . . . . 16
Y (w)(:= X(w) ∩ A). . . . . . . . . . . . . . . .15

  A solution is given to the following problem: how to compute the
multiplicity, or more generally the Hilbert function, at a point on a Schubert
variety in an orthogonal Grassmannian. Standard monomial theory is applied to
translate the problem from geometry to combinatorics. The solution of the
resulting combinatorial problem forms the bulk of the paper. This approach has
been followed earlier to solve the same problem for the Grassmannian and the
symplectic Grassmannian.
  As an application, we present an interpretation of the multiplicity as the
number of non-intersecting lattice paths of a certain kind.
  Taking the Schubert variety to be of a special kind and the point to be the
"identity coset," our problem specializes to a problem about Pfaffian ideals
treatments of which by different methods exist in the literature. Also
available in the literature is a geometric solution when the point is a
"generic singularity."

Introduction 4
Organization of the paper . . . . . . . . . . . . . . . . . . . . . . . . . 5
Important note added . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
I The theorem 7
1 The set up 7
1.1 The statement of the problem . . . . . . . . . . . . . . . . . . . . 7
1.2 Some convenient choices . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Reduction to the case n even . . . . . . . . . . . . . . . . . . . . 8
2 The theorem 9
2.1 Basic notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Two fundamental definitions . . . . . . . . . . . . . . . . . . . . 11
2.2.1 Definition of v-chain . . . . . . . . . . . . . . . . . . . . . 11
2.2.2 Definition of O-domination . . . . . . . . . . . . . . . . . . 11
2.3 The main theorem and its corollary . . . . . . . . . . . . . . . . . 11
II From geometry to combinatorics 13
3 Reduction to combinatorics 13
3.1 Homogeneous co-ordinate ring of the Schubert variety X(w) . . . 13
3.1.1 The line bundle L on Md(V ) . . . . . . . . . . . . . . . . . 13
3.1.2 The section qθ of L . . . . . . . . . . . . . . . . . . . . . . 13
3.1.3 Standard monomial theory for Md(V ) . . . . . . . . . . . . 14
3.2 Co-ordinate rings of affine patches and tangent cones of X(w) . . 14
3.2.1 Standard monomial theory for affine patches . . . . . . . . . 14
3.2.2 Standard monomial theory for tangent cones . . . . . . . . . 16
4 Further reductions 17
4.1 The main propositions . . . . . . . . . . . . . . . . . . . . . . . . 17
4.2 From the main propositions to the main theorem . . . . . . . . . 18
III The proof 21
5 Terminology and notation 21
5.1 Distinguished subsets . . . . . . . . . . . . . . . . . . . . . . . . . 21
5.1.1 Distinguished subsets of N . . . . . . . . . . . . . . . . . . 21
5.1.2 Attaching elements of I(d, 2d) to distinguished subsets of N 21
5.2 The involution # . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5.2.1 The involution # on I(d, 2d) . . . . . . . . . . . . . . . . . 21
5.2.2 The involution # on N and R . . . . . . . . . . . . . . . . 22
5.3 The subset SC attached to a v-chain C . . . . . . . . . . . . . . 22
5.3.1 Vertical and horizontal projections of an element of ON . . 22
5.3.2 The “connection” relation on elements of a v-chain . . . . . 22
5.3.3 The definition of SC . . . . . . . . . . . . . . . . . . . . . 23
5.3.4 The type of an element α of a v-chain C, and the set SC,α 23
6 O-depth 27
6.1 Definition of O-depth . . . . . . . . . . . . . . . . . . . . . . . . 27
6.2 O-depth and depth . . . . . . . . . . . . . . . . . . . . . . . . . . 29
6.3 O-depth and type . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
7 The map Oπ 34
7.1 Description of Oπ . . . . . . . . . . . . . . . . . . . . . . . . . . 34
7.2 Illustration by an example . . . . . . . . . . . . . . . . . . . . . . 35
7.3 A proposition about Sj,j+1 . . . . . . . . . . . . . . . . . . . . . 38
7.4 Proof of Proposition 7.1.1 . . . . . . . . . . . . . . . . . . . . . . 40
7.5 More observations . . . . . . . . . . . . . . . . . . . . . . . . . . 41
8 The map Oφ 42
8.1 Description of Oφ . . . . . . . . . . . . . . . . . . . . . . . . . . 42
8.2 Basic facts about Tw,j,j+1 and T
w,j,j+1 . . . . . . . . . . . . . . . 45
9 Some Lemmas 46
9.1 Lemmas from the Grassmannian case . . . . . . . . . . . . . . . . 46
9.2 Orthogonal analogues of Lemmas of 9.1 . . . . . . . . . . . . . . 49
9.3 Orthogonal analogues of some lemmas in [7] . . . . . . . . . . . . 51
9.4 More lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
10 The Proof 59
10.1 Proof of Proposition 4.1.1 . . . . . . . . . . . . . . . . . . . . . . 59
10.2 Proof that OφOπ = identity . . . . . . . . . . . . . . . . . . . . 62
10.3 Proof that OπOφ = identity . . . . . . . . . . . . . . . . . . . . 63
10.4 Proof of Proposition 4.1.3 . . . . . . . . . . . . . . . . . . . . . . 64
IV An Application 65
11 Multiplicity counts certain paths 65
11.1 Description and illustration . . . . . . . . . . . . . . . . . . . . . 65
11.2 Justification for the interpretation . . . . . . . . . . . . . . . . . 67
References 71
Index of definitions and notation 73
Introduction
In this paper the following problem is solved: given a Schubert variety in an or-
thogonal Grassmannian (by which is meant the variety of isotropic subspaces of
maximum possible dimension of a finite dimensional vector space with a sym-
metric non-degenerate form—see §1 for precise definitions) and an arbitrary
point on the Schubert variety, how to compute the multiplicity, or more gener-
ally the Hilbert function, of the local ring of germs of functions at that point.
In a sense, our solution is but a translation of the problem: we do not give
closed form formulas but alternative combinatorial descriptions. The meaning
of “alternative” will presently become clear.
The same problem for the Grassmannian was treated in [11, 8, 7, 9, 12]
and for the symplectic Grassmannian in [4]. The present paper is a sequel
to [11, 7, 9, 12, 4] and toes the same line as them. In particular, its strategy
is borrowed from them and runs as follows: first translate the problem from
geometry to combinatorics, or, more precisely, apply standard monomial theory
to obtain an initial combinatorial description of the Hilbert function (the earliest
version of the theory capable of handling Schubert varieties in an orthogonal
Grassmannian is to be found in [17]); then transform the initial combinatorial
description to obtain the desired alternative description. But that is easier said
than done.
While the problem makes sense for Schubert varieties of any kind and stan-
dard monomial theory itself is available in great generality [13, 15], the transla-
tion of the problem from geometry to combinatorics has been made—in [14]—
only for “minuscule1 generalized Grassmannians.” Orthogonal Grassmannians
being minuscule, this translation is available to us and we have an initial combi-
natorial description of the Hilbert function. As to the passage from the initial to
the alternative description—and this is where the content of the present paper
lies—neither the end nor the means is clear at the outset.
The first problem then is to find a good alternative description. But how to
measure the worth of an alternative description? The interpretation of multi-
plicity as the number of certain non-intersecting lattice paths (deduced in §11
from our alternative description) seems to testify to the correctness of our al-
ternative description, but we are not sure if there are others that are equally or
more correct.
The proof of the equivalence of the initial and alternative combinatorial de-
scriptions is, unfortunately, a little technically involved. It builds on the details
of the proofs of the corresponding equivalences in the cases of the Grassmannian
and the symplectic Grassmannian. In [10] it is shown that the equivalence in
the case of the Grassmannian is a kind of KRS correspondence, called “bounded
KRS.” The proof there is short and elegant and it would be nice to realise the
main result of the present paper too in a similar spirit as a kind of KRS corre-
spondence.
The initial description is in terms of “standard monomials” and the alterna-
1Symplectic Grassmannians are not minuscule but can be treated as if they were.
tive description in terms of “monomials in roots.” The equivalence of the two
descriptions thus gives a bijective correspondence between standard monomials
and monomials in roots. Roughly—but not actually—the correspondence maps
each standard monomial to its initial term (with respect to a certain monomial
order). Thus it is natural to wonder whether we can compute the initial ideal
of the ideal of the tangent cone to the Schubert variety at the given point. We
believe that this can be done but that it is far more involved and difficult than
the corresponding computation for Grassmannians and symplectic Grassmanni-
ans (the natural set of generators of the ideal of the tangent cone do not form a
Gröbner basis unlike in those cases). If all goes well, the computation will soon
appear [16].
Taking the Schubert variety to be of a special kind and the point to be the
“identity coset,” our problem specializes to a problem about Pfaffian ideals con-
sidered in [5, 2]. On the other side of the spectrum from the identity coset, so to
speak, lie the “generic singularities,” points that are generic in the complement
of the open orbit of the stabiliser of the Schubert variety. For these, a geometric
solution to the problem appears in [1].
Given that our solution of the problem is but a translation, it makes sense
to ask if one can extract more tangible information—closed form formulas for
example—from our alternative description. See the papers quoted in the previ-
ous paragraph and also [3] for some answers in the special cases they consider.
Organization of the paper
The table of contents indicates how the paper is organized. There is a brief
description at the beginning of every subdivision of the contents therein. An
index of definitions and notation is included, for it would otherwise be difficult
to find the meanings of certain words and symbols.
Important note added
The recent article [6] treats some of the questions addressed here and some that
could be addressed by using the main result proved here. It includes:
• an interpretation of the multiplicity similar to ours.
• a closed formula for the multiplicity (as a specialization of a factorial Schur
function), thereby answering the question we raised above.
• a formula for the restriction to the torus fixed point of the equivariant
cohomology class of a Schubert variety.
The approach in [6] is quite different from ours. In fact, it is the opposite of
ours in that it circumvents the lack of results about initial ideals of tangent
cones, while our prime motivation is to remedy the lack. The starting points
in the two approaches are also different: [6] takes off from certain results of
Kostant-Kumar and Arabia on equivariant cohomology, while our launchpad is
standard monomial theory.
The appearance of [6] notwithstanding, our approach is worthwhile, for,
quite apart from the difference in starting points, there is no way, as far as we
can tell, to the Hilbert function via the approach of [6], nor to the initial ideal,
both of which are interesting in their own right.
Part I
The theorem
Definitions are recalled, the problem formulated, and the theorem stated.
1 The set up
In this section, we state the problem to be addressed after recalling the neces-
sary basic definitions, make some choices that are convenient for studying the
problem, and see why it is enough to focus on a particular case of the problem.
1.1 The statement of the problem
Fix an algebraically closed field of characteristic not equal to 2. Fix a vector
space V of finite dimension n over this field and a non-degenerate symmetric
bilinear form 〈 , 〉 on V . Let d be the integer such that either n = 2d or n =
2d+ 1. A linear subspace of V is said to be isotropic if the form 〈 , 〉 vanishes
identically on it. It is elementary to see that an isotropic subspace of V has
dimension at most d and that every isotropic subspace is contained in one of
dimension d. Denote by Md(V )
′ the closed sub-variety of the Grassmannian
of d-dimensional subspaces consisting of the points corresponding to isotropic
subspaces.
The orthogonal group O(V ) of linear automorphisms of V preserving 〈 , 〉
acts transitively on Md(V )
′, for by Witt’s theorem an isometry between sub-
spaces can be lifted to one of the whole vector space. If n is odd the special
orthogonal group SO(V ) (consisting of form preserving linear automorphisms
with trivial determinant) itself acts transitively on Md(V )
′. If n is even the spe-
cial orthogonal group SO(V ) does not act transitively on Md(V )
′, and Md(V )
has two connected components. We define the orthogonal Grassmannian Md(V )
to be Md(V )
′ if n is odd and to be one of the two components of Md(V )
′ if n
is even.
The Schubert varieties of Md(V ) are defined to be the B-orbit closures
in Md(V ) (with canonical reduced scheme structure), where B is a Borel sub-
group of SO(V ). The choice of B is immaterial, for any two of them are con-
jugate. The question that is tackled in this paper is this: given a point on a
Schubert variety in Md(V ), how to compute the multiplicity (and more gen-
erally, the Hilbert function) of the Schubert variety at the given point? The
answers are contained in Theorem 2.3.1 and Corollary 2.3.2. But in order to
make sense of those statements, we need some preparation.
1.2 Some convenient choices
We now make some choices that are convenient for the study of Schubert vari-
eties. For k an integer such that 1 ≤ k ≤ n, set k∗ := n + 1 − k. Fix a basis
e1, . . . , en of V such that
〈ei, ek〉 =
1 if i = k∗
0 otherwise
The advantage of this choice is: the elements of SO(V ) for which each ek is an
eigenvector form a maximal torus, and the elements that are upper triangular
with respect to this basis form a Borel subgroup (a linear transformation is upper
triangular if for each k, 1 ≤ k ≤ n, the image of ek under the transformation
is a linear combination of e1, . . . , ek). We denote this maximal torus and this
Borel subgroup by T and B respectively. Our Schubert varieties will be orbit
closures of this particular Borel subgroup B.
The B-orbits of Md(V )
′ are naturally indexed by its T -fixed points: each
orbit contains one and only one such point. The T -fixed points are evidently of
the form 〈ei1 , . . . , eid〉, where 1 ≤ i1 < . . . < id ≤ n and for each k, 1 ≤ k ≤ d,
there does not exist j, 1 ≤ j ≤ d, such that i∗k = ij—in other words, for each
ℓ, 1 ≤ ℓ ≤ n, such that ℓ 6= ℓ∗, exactly one of ℓ and ℓ∗ appears in {i1, . . . , id};
in addition, if n is odd, then d+ 1 does not appear in {i1, . . . , id}. Denote the
set of such d-element subsets {i1 < . . . < id} by I
n. We thus have a bijective
correspondence between I ′n and the B-orbits of Md(V )
′. Each B-orbit being
irreducible and open in its closure, it follows that B-orbit closures are indexed
by the B-orbits. Thus I ′n is an indexing set for B-orbit closures in Md(V )
Suppose that n is even—it will be shown presently that it is enough to
consider this case. As already observed, Md(V )
′ has two connected components
on each of which SO(V ) acts transitively. The B-orbits belong to one or the
other component accordingly as the parity of the cardinality of the number of
entries bigger than d in the corresponding element of I ′n. We take Md(V ) to
be the component in which these cardinalities are even. We let In denote the
subset of I ′n consisting of elements for which this cardinality is even. Schubert
varieties in Md(V ) are thus indexed by elements of In.
1.3 Reduction to the case n even
We now argue that it is enough to consider the case n even. Suppose that
n is odd. Let ñ := n + 1 and Ṽ be a vector space of dimension ñ with a
non-degenerate symmetric form. Let ẽ1, . . . , ẽen be a basis of Ṽ as in 1.2. Put
e := ẽd+1 and f := ẽd+2. Take λ to be an element of the field such that
λ2 = 1/2. We can take V to be the subspace of Ṽ spanned by the vectors
ẽ1, . . . , ẽd, λe + λf, ẽd+3, . . . , ẽen, and a basis of V to be these vectors in that
order.
There is a natural map from Md+1(Ṽ )
′ to Md(V ): intersecting with V an
isotropic subspace of Ṽ of dimension d + 1 gives an isotropic subspace of V of
dimension d. This map is onto, for every isotropic subspace of Ṽ (and hence
of V ) is contained in an isotropic subspace of Ṽ of dimension d + 1. It is
also elementary to see that the map is two-to-one (essentially because in a two-
dimensional space with a non-degenerate symmetric form there are two isotropic
lines), and that the two points in any fiber lie one in each component (there
is clearly an element in O(Ṽ ) \ SO(Ṽ ) that moves one element of the fiber to
the other, and so if there was an element of SO(Ṽ ) that also moved one point
to the other, the isotropy at the point would not be contained in SO(Ṽ ), a
contradiction).
We therefore get a natural isomorphism between Md+1(Ṽ ) and Md(V ). We
will now show that the B̃-orbits in Md+1(Ṽ ) correspond under the isomorphism
to B-orbits of Md(V ) (we denote by T̃ and B̃ the maximal torus and Borel
subgroups of SO(Ṽ ) as in §1.2). It will then follow that Schubert varieties in
Md+1(Ṽ ) are isomorphic to those in Md(V ) and the purpose of this subsection
will be achieved.
The group SO(V ) can be realized as the subgroup of SO(Ṽ ) consisting of
the elements that fix e − f . The isomorphism Md+1(Ṽ ) ∼= Md(V ) above is
equivariant for SO(V ), and we have T̃ ∩ SO(V ) = T and B̃ ∩ SO(V ) = B. It
should now be clear that the preimages in Md+1(Ṽ ) of two elements in the same
B-orbit of Md(V ) are in the same B̃-orbit: an element of B that moves one to
the other considered as an element of B̃ moves also the preimage of the one to
that of the other.
On the other hand, the preimages of distinct T -fixed points are distinct
T̃ -fixed points, the corresponding map from I ′n to Ien being given as follows:
i = {i1 < . . . < id} 7→
{̃i1, . . . , ĩd, d+ 1} if i ∈ In
{̃i1, . . . , ĩd, d+ 2} if i ∈ I
n \ In
where
ĩk =
ik if ik ≤ d
ik + 1 if ik ≥ d+ 2
(Note that d + 1 never occurs as an entry in any element of I ′n and that the
elements ĩ1, . . . , ĩd, d + 1 (respectively ĩ1, . . . , ĩd, d + 2) are not in increasing
order except in the trivial case i = {1 < . . . < d}.) Given that each B-orbit
has a T -fixed point and that distinct T̃ -fixed points belong to distinct B̃-orbits,
this implies that the preimages of two elements in distinct B-orbits belong to
distinct B̃-orbits, and the proof is over. �
2 The theorem
The purpose of this section is to state the main theorem and its corollary. We
first set down some basic notation and two fundamental definitions needed in
order to state the theorem.
2.1 Basic notation
We keep the terminology and notations of §1.1, 1.2. As observed in §1.3, it is
enough to consider the case n even. So from now on let n = 2d. Recall that,
for an integer k, 1 ≤ k ≤ 2d, k∗ := 2d + 1 − k. As observed in §1.2, Schubert
varieties in Md(V ) are indexed by In.
Since d now determines n, we will henceforth write I(d) instead of In. In
other words, I(d) is the set of d-element of subsets of {1, . . . , 2d} such that
• for each k, 1 ≤ k ≤ 2d, the subset contains exactly one of k, k∗, and
• the number of elements in the subset that exceed d is even.
We write I(d, 2d) for the set of all d-element subsets of {1, . . . , 2d}. There is a
natural partial order on I(d, 2d) and so also on I(d): v = (v1 < . . . < vd) ≤ w =
(w1 < . . . < wd) if and only if v1 ≤ w1, . . . , vd ≤ wd.
Given v ∈ I(d), the corresponding T -fixed point in Md(V ) (namely, the
span of ev1 , . . . , evd) is denoted e
v. Given w ∈ I(d), the corresponding Schubert
variety in Md(V ) (which, by definition, is the closure of the B-orbit of the T -
fixed point ew with canonical reduced scheme structure) is denoted X(w). The
point ev belongs to X(w) if and only if v ≤ w in the partial order just defined.
Since, under the natural action of B on X(w), each point of X(w) is in the
B-orbit of a T -fixed point ev for some v such that v ≤ w, it is enough to focus
attention on such T -fixed points.
For the rest of this section an element v of I(d) will remain fixed.
We will be dealing extensively with ordered pairs (r, c), 1 ≤ r, c ≤ 2d, such
that r is not and c is an entry of v. Let R denote the set of all such ordered
pairs, and set
N := {(r, c) ∈ R | r > c}
OR := {(r, c) ∈ R | r < c∗}
ON := {(r, c) ∈ R | r > c, r < c∗}
= OR ∩N
d := {(r, c) ∈ R | r = c∗}
diagonal
boundary
(r, c)
(c∗, c)
(r, r∗)
The picture shows a drawing of R. We think of r and c in (r, c) as row
index and column index respectively. The columns are indexed from left to
right by the entries of v in ascending order, the rows from top to bottom by
the entries of {1, . . . , 2d} \ v in ascending order. The points of d are those on
the diagonal, the points of OR are those that are (strictly) above the diag-
onal, and the points of N are those that are to the South-West of the poly-
line captioned “boundary of N”—we draw the boundary so that points on
the boundary belong to N. The reader can readily verify that d = 13 and
v = (1, 2, 3, 4, 6, 7, 10, 11, 13, 15, 18, 19, 22) for the particular picture drawn. The
points of ON indicated by solid circles form a v-chain (see §2.2.1 below).
We will be consideringmonomials, also calledmultisets, in some of these sets.
A monomial, as usual, is a subset with each member being allowed a multiplicity
(taking values in the non-negative integers). The degree of a monomial has also
the usual sense: it is the sum of the multiplicities in the monomial over all
elements of the set. The intersection of a monomial in a set with a subset
of the set has also the natural meaning: it is a monomial in the subset, the
multiplicities being those in the original monomial.
We will refer to d as the diagonal.
2.2 Two fundamental definitions
2.2.1 Definition of v-chain
Given two elements (R,C) and (r, c) in ON, we write (R,C) > (r, c) if R > r
and C < c (note that these are strict inequalities). An ordered sequence α, β, . . .
of elements of ON is called a v-chain if α > β > . . . . A v-chain α1 > . . . > αℓ
has head α1, tail αℓ, and length ℓ.
2.2.2 Definition of O-domination
To a v-chain C : α1 > α2 > . . . inON there corresponds, as described in §5.3.3, a
subset SC ofN which, as observed in Proposition 5.3.5, is “distinguished” in the
sense of §5.1.1. To a distinguished subset of N there corresponds, as described
below in §5.1.2, an element of I(d, 2d). Following these correspondences through,
we get an element of I(d, 2d) attached to the v-chain C. Let w(C) denote this
element—sometimes we write wC . (All this makes sense even when C is empty—
w(C) will turn out to be v itself in that case.)
Furthermore, as will be obvious from its definition, the monomial SC is
“symmetric” in the sense of §5.2.2 and contains evenly many elements of the
diagonal d. Thus, by Proposition 5.2.1, the element w(C) of I(d, 2d) belongs
to I(d).
An element w of I(d) is said toO-dominate C if w ≥ w(C), or, equivalently—
and this is important for the proofs—if w dominates in the sense of [7] the mono-
mial SC (for the proof of the equivalence, see [7, Lemma 5.5]). An element w of
I(d) O-dominates a monomial S of ON (repsectively of OR) if it O-dominates
every v-chain in S (respectively in S ∩ON).
2.3 The main theorem and its corollary
Theorem 2.3.1 Fix a positive integer d and elements v ≤ w of I(d). Let V
be a vector space of dimension 2d with a symmetric non-degenerate bilinear
form (over a field of characteristic not 2). Let X(w) be the Schubert variety
corresponding to w in the orthogonal Grassmannian Md(V ), and e
v the torus
fixed point of X(w) corresponding to v. Let Rwv denote the associated graded
ring with respect to the unique maximal ideal of the local ring of germs at ev of
functions on X(w). Then, for any non-negative integer m, the dimension as a
vector space of the homogeneous piece of Rwv of degree m equals the cardinality
of the set Sw(v)(m) of monomials of degree m of OR that are O-dominated
by w.
The proof of this theorem occupies us for most of this paper. It is reduced
in §3, by an application of standard monomial theory, to combinatorics. The
resulting combinatorial problem is solved in §4–10. For now, let us note the
following immediate consequence:
Corollary 2.3.2 The multiplicity at the point ev of the Schubert variety X(w)
equals the number of monomials in ON of maximal cardinality that are square-
free and O-dominated by w.
Proof: The proof of Corollary 2.2 of [7] holds verbatim here too. �
Part II
From geometry to combinatorics
The problem is translated from geometry to combinatorics. The main combi-
natorial results are formulated.
3 Reduction to combinatorics
In this section we translate the problem from geometry to combinatorics. In §3.1
we recall from [17] the theorem that enables the translation. The translation
itself is done in 3.2 and follows [14].
3.1 Homogeneous co-ordinate ring of the Schubert vari-
ety X(w)
3.1.1 The line bundle L on Md(V )
Let Md(V ) ⊆ Gd(V ) →֒ P(∧
dV ) be the Plücker embedding (where Gd(V )
denotes the Grassmannian of all d-dimensional subspaces of V ). The pull-back
toMd(V ) of the line bundle O(1) on P(∧
dV ) is the square of the ample generator
of the Picard group ofMd(V ). Letting L denote the ample generator, we observe
that it is very ample and want to describe the homogeneous coordinate rings
of Md(V ) and its Schubert subvarieties in the embedding defined by L.
3.1.2 The section qθ of L
For θ in I(d, 2d), let pθ denote the corresponding Plücker coordinate. Consider
the affine patch A of P(∧dV ) given by pǫ = 1, where ǫ := (1, . . . , d). The
intersection A ∩Gd(V ) of this patch with the Grassmannian is an affine space.
Indeed the d-plane corresponding to an arbitrary point z of A ∩ Gd(V ) has a
basis consisting of column vectors of a matrix of the form
where I is the identity matrix and A an arbitrary matrix both of size d × d.
The association z 7→ A is bijective. The restriction of a Plücker coordinate pθ
to A∩Gd(V ) is given by the determinant of a submatrix of size d× d of M , the
entries of θ determining the rows to be chosen from M to form the submatrix.
As can be readily verified, a point z of A ∩ Gd(V ) represents an isotropic
subspace if and only if the corresponding matrix A = (aij) is skew-symmetric
with respect to the anti-diagonal : aij + aj∗i∗ = 0, where the columns and rows
of A are numbered 1, . . . , d and d+1, . . . , 2d respectively. For example, if d = 4,
then a matrix that is skew-symmetric with respect to the anti-diagonal looks
like this: 
−d −c −b 0
−g −f 0 b
−i 0 f c
0 i g d
Since the set of these matrices is connected and contains the point that is
spanned by e1, . . . , ed, it follows that A∩Gd(V ) does not intersect the other com-
ponent of Md(V )
′. In other words, pǫ vanishes everywhere on Md(V )
′ \Md(V ).
Now suppose that θ belongs to I(d). Computing pθ/pǫ as a function on the
affine patch pǫ 6= 0, we see that it is the determinant of a skew-symmetric matrix
of even size, and therefore a square. The square root, which is determined up
to sign, is called the Pfaffian. This suggests that pθ itself is a square: more
precisely that there exists a section qθ of the line bundle L on Md(V ) such that
q2θ = pθ. A weight calculation confirms this to be the case. The qθ are also
called Pfaffians.
3.1.3 Standard monomial theory for Md(V )
A standard monomial in I(d) is a totally ordered sequence θ1 ≥ . . . ≥ θt (with
repetitions allowed) of elements of I(d). Such a standard monomial is said to
be w-dominated for w ∈ I(d) if w ≥ θ1. To a standard monomial θ1 ≥ . . . ≥ θt
in I(d) we associate the product qθ1 · · · qθt , where the qθ are the sections defined
above of the line bundle L. Such a product is also called a standard monomial
and it is said to be dominated by w for w ∈ I(d) if the underlying monomial in
I(d) is dominated by w. Standard monomial theory for Md(V ) says:
Theorem 3.1.1 (Seshadri [17]) Standard monomials qθ1 · · · qθr of degree r form
a basis for the space of forms of degree r in the homogeneous coordinate ring of
Md(V ) in the embedding defined by the ample generator L of the Picard group.
More generally, for w ∈ I(d), the w-dominated standard monomials of degree r
form a basis for the space of forms of degree r in the homogeneous coordinate
ring of the Schubert subvariety X(w) of Md(V ).
3.2 Co-ordinate rings of affine patches and tangent cones
of X(w)
From Theorem 3.1.1 one can deduce rather easily, as we now show, bases for
co-ordinate rings of affine patches of the form qv 6= 0 and of tangent cones of
Schubert varieties. An element v of I(d) will remain fixed for the rest of this
section. To simplify notation we will suppress explicit reference to v.
3.2.1 Standard monomial theory for affine patches
Let A denote the affine patch of P(H0(Md(V ), L)
∗) given by qv 6= 0. The
origin of the affine space A is identified as the T -fixed point ev. The functions
fθ := qθ/qv, v 6= θ ∈ I(d), provide a set of coordinate functions on A. Monomials
in these fθ form a k-basis for the polynomial ring k[A] of functions on A, where k
denotes the underlying field.
Fix w ≥ v in I(d), so that the point ev belongs to the Schubert varietyX(w),
and let Y (w) be the affine patch of X(w) defined thus:
Y (w) := X(w) ∩ A.
The coordinate ring k[Y (w)] of Y (w) is a quotient of the polynomial ring k[A],
and the proposition that follows identifies a subset of the monomials in fθ which
forms a k-basis for k[Y (w)].
We say that a standard monomial θ1 ≥ . . . ≥ θt in I(d) is v-compatible if for
each k, 1 ≤ k ≤ t, either θk 
 v or v 
 θk. Given w in I(d), we denote by SM
the set of w-dominated v-compatible standard monomials.
Proposition 3.2.1 As θ1 ≥ . . . ≥ θt runs over the set SM
w of w-dominated
v-compatible standard monomials, the elements fθ1 · · · fθt form a basis for the
coordinate ring k[Y (w)] of the affine patch Y (w) = X(w) ∩ A of the Schubert
variety X(w).
Proof: The proof is similar to the proof of Proposition 3.1 of [7]. First consider
a linear dependence relation among the fθ1 · · · fθt . Replacing fθ by qθ and “ho-
mogenizing” by qv yields a linear dependence relation among the w-dominated
standard monomials qθ1 · · · qθs restricted to X(w), and so the original relation
must only have been the trivial one, for by Theorem 3.1.1 the qθ1 · · · qθs are
linearly independent on X(w).
To prove that fθ1 · · · fθt generate k[Y (w)] as a vector space, we make the
following claim: if qµ1 · · · qµr be any monomial in the Pfaffians qθ, and qτ1 · · · qτs
a standard monomial that occurs with non-zero co-efficient in the expression for
(the restriction to X(w) of) qµ1 · · · qµr as a linear combination of w-dominated
standard monomials, then τ1∪· · ·∪τs = µ1∪· · ·∪µr as multisets of {1, . . . , 2d}.
To prove the claim, consider the maximal torus T of SO(V ) as in §1.2. The affine
patch A is T -stable and there is an action of T on k[Y (w)]. The sections qθ are
eigenvectors for T with corresponding characters ǫθ1+· · ·+ǫθd , where ǫk denotes
the character of T given by the projection to the diagonal entry on row k. The
claim now follows since eigenvectors corresponding to different characters are
linearly independent.
Let fµ1 · · · fµr be an arbitrary monomial in the fθ. Fix an integer h such
that h > r(d − 1) and consider the expression for (the restriction to X(w) of)
qµ1 · · · qµr · q
v as a linear combination of w-dominated standard monomials. We
claim that qv occurs in every standard monomial qτ1 · · · qτr+h in this expression
(from which it will follow that the τj are all comparable to v). Suppose that
none of τ1, . . . , τr+h equals v. For each τj there is at least one entry of v that
does not occur in it. The number of occurrences of entries of v in τ1 ∪ · · · ∪ τr+h
is thus at most (r + h)(d − 1). But these entries occur at least hd times in
µ1 ∪ · · · ∪ µr ∪ v ∪ · · · ∪ v (where v is repeated h times), a contradiction to the
claim proved in the previous paragraph. Hence our claim is proved. Dividing
by qr+hv the expression for qµ1 · · · qµr .q
v as a linear combination of w-dominated
standard monomials provides an expression for fµ1 · · · fµr as a linear combina-
tion of fθ1 · · · fθt , as θ1 ≥ . . . ≥ θt varies over SM
3.2.2 Standard monomial theory for tangent cones
The affine patch Md(V )∩A of the orthogonal Grassmannian Md(V ) is an affine
space whose coordinate ring can be taken to be the polynomial ring in variables
of the form X(r,c) with (r, c) ∈ OR, where (as in §2.1)
OR = {(r, c) | 1 ≤ r, c ≤ 2d, r 6∈ v, c ∈ v, r < c∗}
Taking d = 5 and v = (1, 3, 4, 6, 9) for example, a general element of Md(V )∩A
has a basis consisting of column vectors of a matrix of the following form:

1 0 0 0 0
X21 X23 X24 X26 0
0 1 0 0 0
0 0 1 0 0
X51 X53 X54 0 −X26
0 0 0 1 0
X71 X73 0 −X54 −X24
X81 0 −X73 −X53 −X23
0 0 0 0 1
0 −X81 −X71 −X51 −X21

The expression for fθ = qθ/qv in terms of the X(r,c) is a square root of the
determinant of the submatrix of a matrix like the one above obtained by choosing
the rows given by the entries of θ. Thus fθ is a homogeneous polynomial of
degree the v-degree of θ, where the v-degree of θ is defined as one half of the
cardinality of v \ θ.
Since the ideal of the Schubert variety X(w) in the homogeneous coordinate
ring of Md(V ) is generated
2 by the qτ , τ ∈ I(d) such that τ 6≤ w, it follows
that the ideal of Y (w) := X(w) ∩ A in Md(V ) ∩ A is generated by the the fτ ,
τ ∈ I(d) such that τ 6≤ w. We are interested in the tangent cone to X(w)
at ev (or, what is the same, the tangent cone to Y (w) at the origin), and since
k[Y (w)] is graded, its associated graded ring with respect to the maximal ideal
corresponding to the origin is k[Y (w)] itself.
Proposition 3.2.1 says that the graded piece of k[Y (w)] of degree m is gener-
ated as a k-vector space by elements of degreem of the set SMw of w-dominated
v-compatible standard monomials, where the degree of a standard monomial
θ1 ≥ . . . ≥ θt is defined to be the sum of the v-degrees of θ1, . . . , θt. To prove
Theorem 2.3.1 it therefore suffices to prove the following:
2This is a consequence of Theorem 3.1.1. It is easy to see that the qτ such that τ 6≤ w vanish
on X(w). Since all standard monomials form a basis for the homogeneous coordinate ring
of Md(V ) in P(H
0(Md(V ), L)
∗), it follows that w-dominated standard monomials span the
quotient ring by the ideal generated by such qτ . Since such monomials are linearly independent
in the homogeneous coordinate ring of X(w), the desired result follows.
Theorem 3.2.2 The set SMw(m) of standard monomials in I(d) of degree m
that are w-dominated and v-compatible is in bijection with the set Sw(v)(m) of
monomials in OR of degree m that are O-dominated by w.
4 Further reductions
In the last section, we reduced the proof of our main theorem (Theorem 2.3.1)
to that of Theorem 3.2.2. We now reduce the proof of Theorem 3.2.2 to that of
Propositions 4.1.1, 4.1.2 and 4.1.3 below. These propositions will eventually be
proved in §10.
4.1 The main propositions
Fix once and for all an element v of I(d). The bijection stated in Theorem 3.2.2
will be described by means of two maps Oπ and Oφ whose definitions will be
given in §7 and §8 below. We will now state some properties of these maps.
In §4.2 we will see how Theorem 3.2.2 follows once these properties are estab-
lished.
The map Oπ associates to a monomial S in ON a pair (w,S′) consisting
of an element w of I(d) and a “smaller” monomial S′ in ON. This map enjoys
the following good properties:
Proposition 4.1.1 1. w ≥ v.
2. v-degree(w) + degree(S′) = degree(S).
3. w O-dominates S′.
4. w is the least element of I(d) that O-dominates S.
The map Oφ, on the other hand, associates a monomial in ON to a pair (w,T)
consisting of an element w of I(d) with w ≥ v and a monomial T in ON that is
O-dominated by w.
Proposition 4.1.2 The maps Oπ and Oφ are inverses of each other.
For an integer f , 1 ≤ f ≤ 2d, consider the following conditions, the first on
a monomial S in ON, the second on an element w of I(d):
(‡) f is not the row index of any element of S and f⋆ is not the
column index of any element of S.
(‡) f is not an entry of w.
(It is convenient to the use the same notation (‡) for both conditions.)
Proposition 4.1.3 Assume that v satisfies (‡)—all references to (‡) in this
proposition are with respect to a fixed f , 1 ≤ f ≤ 2d.
1. Let w be an element of I(d) with w ≥ v and T a monomial in ON that is
O-dominated by w. If w and T both satisfy (‡), then so does Oφ(w,T).
2. If a monomial S in ON satisfies (‡), then so do the “components” w and
S′ of its image under Oπ.
4.2 From the main propositions to the main theorem
Let us now see how Theorem 3.2.2 follows from the propositions of §4.1. Most
of the following argument runs parallel to its counterparts in the case of the
Grassmannian and symplectic Grassmannian (Propositions 4.1.1 and 4.1.2 have
their counterparts in [7, 4]), but, in the case that d is odd, the part involving
the “mirror image” requires additional work. This is where Proposition 4.1.3
comes in.
Let S, T , and U , denote respectively the sets of monomials in OR, ON, and
OR\ON. Let SMv denote the set of v-compatible standard monomials that are
“anti-dominated” by v: a standard monomial θ1 ≥ . . . ≥ θt is anti-dominated
by v if θt ≥ v (we can also write θt > v since θt 6= v by v-compatibility).
Define the domination map from T to I(d) by sending a monomial in ON to
the least element that O-dominates it. Define the domination map from SMv
to I(d) by sending θ1 ≥ . . . ≥ θt to θ1. Both these maps take, by definition, the
value v on the empty monomial.
Notation 4.2.1 In the following, we use subscripts, superscripts, suffixes, and
combinations thereof to modify the meanings of S, T , U , SM , and SMv.
• superscript: this will be an element w of I(d); when used on T it denotes
O-domination (more precisely, Tw denotes the subset of T consisting of
those elements that are O-dominated by w); when used on SM or SMv
it denotes domination by w.
• subscript: denotes anti-domination (applied only to standard monomials).
• suffix “(m)”: indicates degree (for example, SMwv (m) denotes the set of v-
compatible standard monomials that are anti-dominated by v, dominated
by w, and of degree m).
Repeated application of Oπ gives a map from T to SMv that commutes
with domination (as just defined) and preserves degree. Repeated application
of Oφ gives a map from SMv to T . These two maps being inverses of each
other (Proposition 4.1.2) and so we have a bijection between SMv and T . In
fact, since domination and degree are respected (Proposition 4.1.1), we get a
bijection SMwv (m)
∼= Tw(m).
As explained below, the “mirror image” of the bijection SMv(m) ∼= T (m)
gives a bijection SMv(m) ∼= U(m). Putting these bijections together, we get
the desired result:
SMw(m) =
SMwv (k)× SM
v(m− k)
Tw(k)× U(m− k) = Sw(m).
We now explain how to realize the bijection SMv(m) ∼= U(m) as the “mirror
image” of the bijection SMv(m) ∼= T (m). For an element u of I(d), define
u∗ := (u∗d, . . . , u
1). In the case d is even, the association u 7→ u
∗ is an order
reversing involution, and the argument in [4] for the symplectic Grassmannian
holds here too. In the case d is odd, u∗ is not an element of I(d), and so some
additional work is required.
Recall that a “base element” v of I(d) has been fixed and that our notation
does not explicitly indicate this dependence upon v: for example, OR is depen-
dent upon v. For a brief while now (until the end of this section) we need to
simultaneously handle several base elements of I(d). We will use the following
convention: when the base element of I(d) is not v, we will explicitly indicate
it by means of a suffix. For instance, SM(v∗) denotes the set of v∗-compatible
standard monomials in I(d).
Let us first do the case when d is even. We get a bijection SMv ∼= SMv∗(v
by associating to θ1 ≥ . . . ≥ θt the element θ
t ≥ . . . ≥ θ
1 . The sum of the
v-degrees of θ1, . . . , θt equals the sum of the v
∗-degrees of θ∗t , . . . , θ
1 , so that we
get a bijection SMv(m) ∼= SMv∗(v
∗)(m).
For an element (r, c) of ON(v∗), consider its flip (c, r). Since v belongs
to I(d), the complement of v∗ in {1, . . . , 2d} is v, and it follows that (c, r)
belongs to OR \ ON. This induces a degree preserving bijection T (v∗) ∼= U .
Putting this together with the bijection of the previous paragraph and the one
deduced earlier in this section (using Oπ and Oφ), we get what we want:
SMv(m) ∼= SMv∗(v
∗)(m) ∼= T (v
∗)(m) ∼= U(m).
Now suppose that d is odd. Then the map x 7→ x∗ does not map I(d) to
I(d) but to I(d)∗ (defined as the set consisting of those elements u of I(d, 2d)
such that, for each k, 1 ≤ k ≤ 2d, exactly one of k, k∗ belongs to u, and the
number of entries of u greater than d is odd). We define a map u 7→ ũ from
I(d)∗ to I(d + 1) as follows: ũ := {ũ1, . . . , ũd, d + 2} (the elements are not in
increasing order except in the trivial case u = (1, . . . , d)), where, for an integer
e, 1 ≤ e ≤ 2d, we set
ẽ :=
e if 1 ≤ e ≤ d
e+ 2 if d+ 1 ≤ e ≤ 2d
This map u 7→ ũ is an order preserving injection.
Consider the composition x 7→ x∗ 7→ x̃∗ from I(d) to I(d+1). This is an order
reversing injection. The induced map on standard monomials is an injection
from SMv to SMfv∗(ṽ
∗). It is readily seen that the image under this map
is the subset SMfv∗(ṽ
∗)(‡) consisting of those standard monomials all of whose
elements satisfy (‡) with f = d+1. We have already established (using the maps
Oπ and Oφ) a bijection SMfv∗(ṽ
∗) ∼= T (ṽ∗). It follows from Proposition 4.1.3
that under this bijection the subset SMfv∗(ṽ
∗)(‡) maps to T (ṽ∗)(‡) (defined as
the set of those monomials in ON(ṽ∗) satisfying (‡) with f = d+ 1).
Now T (ṽ∗)(‡) is in degree preserving bijection with U : every element of
degree 1 of T (ṽ∗)(‡) is uniquely of the form (c̃, r̃) for (r, c) in OR\ON, and the
desired bijection is induced from this. Putting all of these together, we finally
SMv ∼= SMfv∗(ṽ
∗)(‡) ∼= T (ṽ∗)(‡) ∼= U.
Thus, in order to prove our main theorem (Theorem 2.3.1), it suffices to
describe the maps Oπ and Oφ and to prove Propositions 4.1.1–4.1.3.
Part III
The proof
The main combinatorial results formulated in §4.1 are proved. An attempt is
made to maintain parallelism with the proofs in [7].
5 Terminology and notation
5.1 Distinguished subsets
5.1.1 Distinguished subsets of N
Following [7, §4], we define a multiset S of N to be distinguished , if, first of all,
it is a subset in the usual sense (in other words, it is “multiplicity free”), and
if, for any two distinct elements (R,C) and (r, c) of S, the following conditions
are satisfied:
A. R 6= r and C 6= c.
B. If R > r, then either r < C or C < c.
In terms of pictures, condition A says that (r, c) cannot lie exactly due North
or East of (R,C) (or the other way around); so we can assume, interchanging
the two points if necessary, that (r, c) lies strictly to the Northeast or Northwest
of (R,C); condition B now says that, if (r, c) lies to the Northwest of (R,C),
then the point that is simultaneously due North of (R,C) and due East of (r, c)
(namely (r, C)) does not belong to N.
5.1.2 Attaching elements of I(d, 2d) to distinguished subsets of N
To a distinguished subset S of N there is naturally associated an element w of
I(d, 2d) as follows: start with v, remove all members of v which appear as column
indices of elements of S, and add row indices of all elements of S. As observed
in [7, Proposition 4.3], this association gives a bijection between distinguished
subsets of N and elements w ≥ v of I(d, 2d). The unique distinguished subset
of N corresponding to an element w ≥ v of I(d, 2d) is denoted Sw.
5.2 The involution #
5.2.1 The involution # on I(d, 2d)
There are two natural order reversing involutions on I(d, 2d). First there is
w 7→ w∗ induced by the natural order reversing involution j 7→ j∗ on {1, . . . , 2d}:
here w∗ has the obvious meaning, namely, it consists of all j∗ such that j
belongs to w. Then there is the map taking w to its complement {1, . . . , 2d}\w.
These two involutions commute. Composing the two we get an order preserving
involution on I(d, 2d) which we denote by w 7→ w#. The elements of the
subset I(d) are fixed points under this involution (there are points not in I(d)
that are also fixed).
5.2.2 The involution # on N and R
For α = (r, c) in N, or more generally in R, define α# = (c∗, r∗). The involution
α 7→ α# is just the reflection with respect to the diagonal d. For a subset or
even multiset S of N (or R), the symbol S# has the obvious meaning. We call
S symmetric if S = S#.
Proposition 5.2.1 An element w ≥ v of I(d, 2d) belongs to I(d) if and only
if the distinguished subset Sw of N corresponding to it as described in §5.1.2 is
symmetric and has evenly many diagonal elements.
Proof: That the symmetry of Sw is equivalent to the condition that w = w
is proved in [4, Proposition 5.7]. Now suppose that Sw is symmetric. We claim
that for an element (r, c) of Sw that is not on the diagonal, either both r and c
are bigger than d or both are less than d + 1. It is enough to prove the claim,
for w is obtained from v by removing the column indices and adding the row
indices of elements of Sw, and it would follow that the number of entries in w
that are bigger than d equals the number of such entries in v plus the number
of diagonal elements in Sw.
We now prove the claim. Since Sw is symmetric, it follows that (c
∗, r∗) also
belongs to Sw. Since Sw is distinguished, it follows that in case r < c
∗ (that
is, if (r, c) lies above the diagonal), we have r < r∗, and so c < r < r∗; and in
case r > c∗, we have c∗ < c, and so c∗ < c < r. Thus the claim is proved. �
5.3 The subset SC attached to a v-chain C
5.3.1 Vertical and horizontal projections of an element of ON
For α = (r, c) in ON (or more generally in OR), the elements pv(α) := (c
∗, c)
and ph(α) := (r, r
∗) of the diagonal d are called respectively the vertical and
horizontal projections of α. In terms of pictures, the vertical projection is the
element of the diagonal due South of α; the horizontal projection is the element
of the diagonal due East of α. The vertical line joining α to its vertical projection
pv(α) and the horizontal line joining α to its horizontal projection ph(α) are
called the legs of α.
5.3.2 The “connection” relation on elements of a v-chain
Let C : α1 = (r1, c1) > α2 = (r2, c2) > · · · be a v-chain in ON. Two consecutive
elements αj and αj+1 of C are said to be connected if the following conditions
are both satisfied:
• their legs are “intertwined”; equivalently and more precisely, this means
that r∗j ≥ cj+1, or, what amounts to the same, rj ≤ c
• the point (rj+1, r
j ) belongs to N; this just means that rj+1 > r
Consider the coarsest equivalence relation on the elements of C generated by
the above relation. The equivalence classes of C with respect to this equivalence
relation are called the connected components of the v-chain C.
This definition has its quirks:
The v-chain C : α > β > γ in the pic-
ture has {α, β} and {γ} as its connected
components; but the “sub” v-chain α > γ
of C is connected (as a v-chain in its own
right).
diagonal
boundary
5.3.3 The definition of SC
We will define SC as a multiset of N. It is easy to see and in any case stated
explicitly as part of Corollary 5.3.5 that it is multiplicity free and so is actually
a subset of N.
First suppose that C : α1 = (r1, c1) > · · · > αℓ = (rℓ, cℓ) is a connected
v-chain in ON. Observe that, if there is at all an integer j, 1 ≤ j ≤ ℓ, such that
the horizontal projection ph(αj) does not belong to N, then j = ℓ. Define
SC :=
{pv(α1), . . . , pv(αℓ)} if ℓ is even
{pv(α1), . . . , pv(αℓ)} ∪ {ph(αℓ)} if ℓ is odd and ph(αℓ) ∈ N
{pv(α1), . . . , pv(αℓ−1)} ∪ {αℓ, α
} if ℓ is odd and ph(αℓ) 6∈ N
For a v-chain C that is not necessarily connected, let C = C1 ∪ C2 ∪ · · · be
the partition of C into its connected components, and set
SC := SC1 ∪SC2 ∪ · · ·
5.3.4 The type of an element α of a v-chain C, and the set SC,α
We introduce some terminology and notation. Their usefulness may not be
immediately apparent.
Suppose that C : α1 > · · · > αℓ is a connected v-chain. We define the type
in C of an element αj , 1 ≤ j ≤ ℓ, of C to be V, H, or S, accordingly as:
V: j 6= ℓ, or j = ℓ and ℓ is even.
H: j = ℓ, ℓ is odd, and ph(αℓ) ∈ N.
S: j = ℓ, ℓ is odd, and ph(αℓ) 6∈ N.
The type of an element in a v-chain that is not necessarily connected is defined
to be its type in its connected component.
The set SC,α of elements of N generated by an element α of C is defined to
SC,α :=
{pv(α)} if α is of type V in C;
{pv(α), ph(α)} if α is of type H in C;
{α, α#} if α is of type S in C;
Observe that, for a v-chain C, the monomial SC defined in §5.3.3 is the union,
over all elements α of C, of SC,α.
For an element α of a v-chain C, we define qC,α to be pv(α) if α is of type V
or H and to be α if it is of type S.
If the horizontal projection of an element in a v-chain does not belong to N,
then clearly the same is true for every succeeding element. The first such element
of a v-chain is called the critical element.
Proposition 5.3.1 1. The cardinality is odd of a connected component that
has an element of type H or S. Conversely, if the cardinality of a compo-
nent is odd, then it has an element of type H or S.
2. An element of type H or S can only be the last element in its connected
component.
3. The critical element has type either V or S. No element before it can be of
type S and every element after it is of type S. In particular, any element
that succeeds an element of type S is of type S.
Proof: Clear from definitions. �
Proposition 5.3.2 Let α > γ be elements of a v-chain C (we are not assuming
that they are consecutive).
1. If α > γ is connected as a v-chain in its own right, then α is connected to
its next member in C; that is, α cannot be the last element in its connected
component in C.
2. If α > γ is not connected as a v-chain in its own right and the legs of α and
γ intertwine, then the connected component of γ in C is the singleton {γ},
and γ has type S in C.
Proof: Clear from definitions. �
Proposition 5.3.3 Let E : α > . . . > ζ be a v-chain, D and D′ two v-chains
with tail α, and C, C′ the concatenations of D, D′ respectively with E. Then
1. The last element in the connected component containing α is the same
in C and C′ (and this is the same as in E).
Let λ denote this element.
2. The only element among α, . . . , ζ that possibly has different types in C
and C′ is λ.
Proof: (1): Whether or not two successive elements in a v-chain are connected
is independent of other elements in the v-chain.
(2): The type of an element in a v-chain is V unless it is the last element
in its connected component. And the type of the last element in a component
depends on the cardinality of the component. The components of E not contain-
ing α are still components in C and C′. In contrast, the component containing α
could possibly be larger in C (respectively C′) and hence its cardinality could
be different. �
For an element α = (r, c) of N, we define α(up) to be α itself if α is either on
or above the diagonal d (more precisely, if r ≤ c∗), and to be its “reflection” in
the diagonal (more precisely, (c∗, r∗)) if α is below the diagonal (more precisely,
if r > c∗). For a monomial S of N, S(up) is defined to be the intersection
of S (as a multiset) with the subset ON ∪ d of N. The notations α(down) and
S(down) have similar meanings.
Caution: It is not true that S(up) = {α(up)|α ∈ S} (in the obvious sense one
would make of the right hand side). In particular, for a singleton monomial {α},
it is not always true that {α}(up) = {α(up)}.
Proposition 5.3.4 Let α and β be elements of a v-chain C. Let us use α′ and
β′ respectively to denote elements of SC,α(up) and SC,β(up).
1. If α > β (these elements are not necessarily consecutive in C), then, given
β′, there exists α′ such that α′ > β′. In fact, this is true for every choice
of α′ except when
(*) α is of type H, and ph(α) 6> β
′ for some β′ ∈ SC,β.
In particular, qC,α > β
′ and qC,α > qC,β.
2. Conversely, suppose that α′ > β′ for some choice of α′ and β′. Then
α ≥ β; if equality occurs, then α is of type H, α′ = pv(α) and β
′ = ph(α).
In particular, if α′ > qC,β (or more specially qC,α > qC,β), then α > β.
3. If (*) holds for α > β in C, then
(a) the critical element of C is the one just after α; in particular, α is
uniquely determined.
(b) all elements of C succeeding α are of type S; in particular, β is of
type S and β′ = β.
(c) (*) holds for γ in place of β for every γ in C that succeeds α.
Proof: (1) If α is of type V or H, we need only take α′ = pv(α), for pv(α) >
pv(β), pv(α) > ph(β), and pv(α) > β. Now suppose that α is of type S. Then β
too is of type S (Proposition 5.3.1 (3)), so β′ can only be β, and the first part
of (1) is proved.
It follows from the above that if α′ = pv(α) or if α has type S, then α
′ > β′
independent of the choice of α′. So if α′ 6> β′, then (*) holds and α′ = ph(α).
(3) Let λ be the immediate successor of α in C. Then α is not connected
to λ (Proposition 5.3.1 (2)). Since ph(α) 6> β
′, it follows that α and β have
intertwining legs. Therefore so do α and λ. By Proposition 5.3.2 (2), λ has
type S in C.
Since α has type H and λ type S, it follows immediately from the definition of
the critical element that λ is the critical element. This proves (a). Assertion (b)
now follows from Proposition 5.3.1 (3). For (c), write ph(α) = (a, a
∗), λ =
(R,C), and γ = (r, c). Then R < a∗, for α and λ have intertwining legs but
are not connected. So c < r ≤ R < a∗. This means ph(α) 6> γ. And γ being of
type S (by (b)), we can take γ′ = γ.
(2) Suppose that α 6≥ β. Then β > α. By the second part of (1) above, β
is of type H and β′ = ph(β); by item (b) of (3), α is of type S, so α
′ = α. This
leads to the contradiction β > α > ph(β). �
Corollary 5.3.5 The multiset SC attached to a v-chain C is a distinguished
subset of N in the sense of 5.1.1.
Proof: If α in C is of type V or S, then SC,α is a singleton; if it is of type H,
then SC,α = {pv(α), ph(β)}. So there can be no violation of conditions A and B
of §5.1.1 by elements of SC,α.
Suppose α > β. By Proposition 5.3.4 (1), we have α′ > β′ for any choice
of α′ ∈ SC,α and β
′ ∈ SC,β except when the condition (*) holds. By (3)
of the same proposition, if (*) holds, then β′ = β, and writing β = (r, c),
ph(α) = (a, a
∗), we have r < a (since α > β) and c < r < a∗ (see proof of
item 3(c) of the proposition). Thus there can be no violation of conditions A
and B of §5.1.1. �
Corollary 5.3.6 Let S be a v-chain in ON and w an element of I(d). If w
O-dominates S, then w dominates in the sense of [7] the monomial S ∪ S#
of N.
Proof: By [4, Proposition 5.15], it is enough to show that w dominates S. Let
C : α1 > . . . > αt be a v-chain in S. Writing αj = (rj , cj) and qC,αj = (Rj , Cj)
we have rj ≤ Rj and Cj ≤ cj . By Proposition 5.3.4 (1), we have qC,α1 > . . . >
qC,αt . Since w O-dominates S, it in particular dominates qC,α1 > . . . > qC,αt
and so also C. �
6 O-depth
The concept of O-depth defined in §6.1 below plays a key role in this paper. As
the name suggests, it is the orthogonal analogue of the concept of depth of [7].
In §6.2 below, it is observed that the O-depth is no smaller than depth in the
sense of [7]. In §6.3, some observations about the relation between O-depths
and types of elements in v-chains are recorded.
6.1 Definition of O-depth
The O-depth of an element α in a v-chain C in ON is the depth in SC in the
sense of [7] of qC,α: in other words, it is the depth in SC of pv(α) in case α is of
type V or H, and of α (equivalently of α#) in case α is of type S. It is denoted
O-depthC(α). The O-depth of an element α in a monomial S of ON is the
maximum, over all v-chains C in S containing α, of the O-depth of α in C. It
is denoted O-depth
(α). Finally, the O-depth of a monomial S in ON is the
maximum of the O-depths in S of all the elements of S.
There is a conflict in the above definitions: Is the O-depth of an element
of a v-chain C the same as its depth as an element of the monomial C? In
other words, could the O-depth of an element in a v-chain be exceeded by its
O-depth in a sub-chain? The conflict is resolved by the first item of the following
proposition.
Proposition 6.1.1 1. For v-chains C ⊆ D, the O-depth in C of an element
of C is no more than its O-depth in D.
2. If a v-chain C is an initial segment of a v-chain D, then the O-depths
in C and D of an element of C are the same.
Proof: (1): By an induction on the difference in the cardinalities of D and C,
we may assume that D has one more element than C. Call this extra element δ.
Suppose that δ lies between successive elements α and β of C (the modifications
needed to cover the extreme cases when it goes at the beginning or the end are
being left to the reader).
The only elements of C that could possibly undergo changes of type on
addition of δ are α and the last element in the connected component of β,
which let us call β′. If there are no type changes, then SC ⊆ SD and the
assertion is immediate. The only type change that α can undergo is from H
to V. The type changes that β′ can undergo are: H to V; V to H; S to V; V to
S. An easy enumeration of cases shows that only one of α and β′ can undergo
a type change.
We need not worry about changes from V to H for in this case SC ⊆ SD.
First let us suppose that α undergoes a change of type (from H to V). Then
δ is connected to α. It follows from Proposition 5.3.1 (1) that δ has type V in D:
the connected component of α in C has odd number of elements, so if δ happens
to be the last element in its connected component in D, the number of elements
in that component will be even. Replacing an occurrence of ph(α) in a v-chain
of SC by pv(δ) would result in a v-chain in SD (by Proposition 5.3.4 (1)), and
this case is settled.
Now suppose that β′ undergoes a type change. Then δ is connected to β and
δ is of type V in D (Proposition 5.3.1 (2)). Replacing by pv(δ) any occurrence
in a v-chain in SC of pv(β
′), ph(β
′), β′ accordingly as the type of β′ in C
is V, H, or S, (not necessarily in the same place but at an appropriate place)
would result in a v-chain in SD (by Proposition 5.3.4 (1)), and we see that the
O-depth cannot decrease.
(2): It follows from Proposition 5.3.4 (2) that, for an element α of C, con-
tributions to SD from elements beyond α (in particular from those not in C)
do not affect the depth in SD of qD,α. Looking for the possibility of differences
in types in C and D of elements of C, we see that the only element of C that
has possibly a different type in D is its last element. And this too can change
type only from H to V.
The above two observations imply that the calculations of O-depths in C
and D of an element α of C are no different: we would be considering the depth
in SC and SD respectively of the same element (either pv(α) or α), and the
differences in SD and SC have no effect on this consideration. �
Corollary 6.1.2 If C ⊆ D are v-chains in ON, then wC ≤ wD (although it is
not always true that SC ⊆ SD).
Proof: By [7, Lemma 5.5], it is enough to show that every v-chain in SC is
dominated by wD. Let β1 = (r1, c1) > · · · > βt = (rt, ct) be an arbitrary v-chain
in SC . To show that it is dominated by wD, it is enough, by [7, Lemma 4.5], to
show the existence of a v-chain (R1, C1) > · · · > (Rt, Ct) in SD with rj ≤ Rj
and Cj ≤ cj for 1 ≤ j ≤ t. Such a v-chain exists by the proof of (1) of Proposi-
tion 6.1.1. �
Corollary 6.1.3 1. Let S be a monomial in ON and α ∈ S. Then there ex-
ists a v-chain C in S with tail α such that O-depth
(α) = O-depthC(α).
2. For elements α > γ in a v-chain C (these need not be consecutive), we
have O-depthC(α) < O-depthC(γ).
3. For elements α > γ of a monomial S in ON, we have O-depth
(α) <
O-depth
4. No two elements of the same O-depth in a monomial in ON are compa-
rable.
Proof: (1) This follows from (2) of the Proposition above and the definition
of O-depth.
(2) This follows from Proposition 5.3.4 (1) and the definition of O-depth.
(3) By (1), there exists a v-chain C with tail α such that O-depth
(α) =
O-depthC(α). Concatenate C with α > γ and letD denote the resulting v-chain.
By (2) of the Proposition above, O-depthC(α) = O-depthD(α). By (2) above,
O-depthD(α) < O-depthD(γ). And finally, O-depthD(γ) ≤ O-depthS(γ) by
the definition of O-depth
(4) Immediate from (3). �
Corollary 6.1.4 Let β > γ be elements of a v-chain C of elements of ON. Let
E be a v-chain in SC with tail qC,γ and length O-depthC(γ). Then qC,β occurs
in E.
Proof: It is enough to show that for α′ 6= qC,β in E, either α
′ > qC,β or
qC,β > α
′. Let α be in C such that qC,β 6= α
′ ∈ SC,α. If β ≥ α, then qC,β > α
by Proposition 5.3.4 (1). If α > β and α′ 6> qC,β , then, by (1) and (3) of the
same proposition, α′ 6> qC,γ , a contradiction. �
6.2 O-depth and depth
Lemma 6.2.1 The O-depth of an element α in a monomial S of ON is no
less than its depth (in the sense of [7]) in S ∪S#.
Proof: Let C : α1 > . . . > αt be a v-chain in S∪S
# with tail αt = α, where t
is the depth of α in S ∪S#. We then have α1(up) > . . . > αt(up), so we may
assume C to be in S. By Proposition 5.3.4 (1), qC,α1 > . . . > qC,αt in SC . So
depth
S∪S#(α) = t ≤ depthSC (qC,αt) ≤ O-depthS(α). �
6.3 O-depth and type
We begin by defining some useful terminology. Let (r, c) and (R,C) be two
elements of R. To say that (R,C) dominates (r, c) means that r ≤ R and C ≤ c
(in terms of pictures, (r, c) lies (not necessarily strictly) to the Northeast of
(R,C)). To say that they are comparable means that either (R,C) > (r, c) or
(r, c) > (R,C). While this is admittedly strange, there will arise no occasion
for confusion.
For an integer i, we let i(odd) be the largest odd integer not bigger than i
and i(even) the smallest even integer not smaller than i.
Lemma 6.3.1 1. For consecutive elements α > β of a v-chain C,
O-depthC(β) =
O-depthC(α) + 2 if and only if α is of
type H and ph(α) > β
O-depthC(α) + 1 otherwise
2. For an element of a v-chain C such that either its horizontal projection
belongs to N or it is connected to its predecessor, the parity of its O-depth
in C is the same as that of its ordinality in its connected component in C.
3. The O-depth in a v-chain of an element of type H is odd.
4. If in a v-chain an element of type V is the last in its connected component,
then its O-depth is even.
5. If in a v-chain C there is an element of O-depth d, then
(a) for every odd integer d′ not exceeding d, there is in C an element of
O-depth d′.
(b) if, for an even integer d′ not exceeding d, there is no element in C of
O-depth d′, then the element α in C of O-depth d′ − 1 is of type H,
and ph(α) > β, where β denotes the immediate successor of α in C.
6. Let C be a v-chain and α an element of type H in C. Then the depth in
SC of ph(α) equals O-depthC(α) + 1. In particular, this depth is even.
Proof: (1): From items 1 and 3(a) of Proposition 5.3.4, it follows that, for
γ in C with γ > α, if γ′ 6> qC,α for some γ
′ in SC,γ , then γ
′ 6> qC,β. Thus
O-depthC(β) exceeds O-depthC(α) by the number of elements in SC,α that
dominate qC,β . This number is 1 if α is of type V, or of type S, or of type H
and ph(α) 6> β; it is 2 if α is of type H and ph(α) > β (note that ph(α) > β if
and only if ph(α) > qC,β).
(2): Let λ be such an element. Everything preceding λ in C is of type H
or V (Proposition 5.3.1 (3)). Let λ belong to the kth connected component,
and n1, . . . , nk be respectively the cardinalities of the first, . . . , k
th connected
components. By (1) above and item 3(b) of Proposition 5.3.4, O-depthC(λ)
is n1(even) + · · · + nk−1(even) plus the ordinality of λ in the k
th connected
component.
(3) and (4): These are special cases of 2.
(5): This follows easily from (1) and (3).
(6): It follows from Proposition 5.3.4 (2) that there is no element γ in SC
that lies between pv(α) and ph(α) (meaning pv(α) > γ > ph(α)), so the asser-
tion holds. �
Corollary 6.3.2 For a v-chain C in ON, if the O-depths of elements in C are
bounded by k, then the depths of elements in SC are bounded by k(even).
Proof: The depth of qC,α in SC for any α in C is at most k by hypothesis.
An element of SC that is not qC,α for any α in C can only be of the form
ph(α) for some α. By Proposition 5.3.4, depthSCpv(α) = depthSCph(α) − 1,
which implies depth
ph(α) ≤ k + 1. If, moreover, k is even, then by (3) of
Lemma 6.3.1 depth
ph(α) = depthSCpv(α) + 1 ≤ (k − 1) + 1 = k. �
Proposition 6.3.3 Given a monomial S in ON and an element α in it, there
exists a v-chain C in S with tail α such that O-depthC(β) = O-depthS(β) for
every β in C.
Proof: Proceed by induction on d := O-depth
(α). Choose a v-chain D in S
with tail α such that O-depthD(α) = O-depthS(α) (such a v-chain exists by
Corollary 6.1.3 (1)). Let α′ be the element in D just before α. It follows from
item (3) of Corollary 6.1.3 and item (1) of Lemma 6.3.1 that O-depth
(α′) (as
also O-depthD(α
′)) is either d − 1 or d − 2. By induction, there exists a v-
chain C′ with tail α′ that has the desired property. Let C be the concatenation
of C′ with α′ > α.
We claim that C has the desired property. The only thing to be proved is
that O-depthC(α) = d. By item (1) of Lemma 6.3.1, we have O-depthC(α) ≥
O-depthC′(α
′) + 1. In particular, the claim is proved in case O-depthC′(α
is d − 1, so let us assume that O-depthC′(α
′) is d− 2. It now follows from the
same item that α′ has type H in D and ph(α
′) > α; it further follows that it is
enough to show that α′ has type H in C.
Since α′ has type H in D, it follows (from item (2) of Proposition 5.3.1) that
α′ > α is not connected and (from item (3) of Lemma 6.3.1) that d− 2 is odd.
Now, by item (4) of Lemma 6.3.1, the type in C′ of α′ cannot be V, so it is H,
and the claim is proved. �
Corollary 6.3.4 Let S be a monomial in ON, β an element of S, and i an
integer such that i < O-depth
(β). Then
(a) If i is odd, there exists an element α in S of O-depth i such that α > β.
(b) If i is even and there is no element α in S of O-depth i such that α > β,
then there is element α in S of O-depth i− 1 such that ph(α) > β.
Proof: Choose a v-chain C in S having tail β and the good property of Propo-
sition 6.3.3. Apply Lemma 6.3.1 (5). �
Corollary 6.3.5 Let C be a v-chain in ON with tail α such that O-depthC(α)
is odd. Let A be a v-chain in ON with head α, and D the concatenation of C
with A. Let C′ denote the v-chain C \ {α}. Then
1. The type of an element of A is the same in both A and D. In particular,
SA ⊆ SD and qA,β = qD,β for β in A.
2. The type of an element of C′ is the same in both C′ and D. In particular,
SC′ ⊆ SD.
3. SD = SC′ ∪ SA (disjoint union); letting j0 := O-depthC(α) we have
j0 = SA and (SD)1 ∪ · · · ∪ (SD)j0−1 = SC′ . (For a monomial S,
the subset of elements of depth at least i is denoted Si, and the subset of
elements of depth exactly i is denoted Si.)
Proof: (1) Generally (meaning without the assumption that O-depthC(α) is
odd), the only element of A that could possibly have a different type in D is
the last one in the first connected component of A; whether or not it changes
type depends exactly upon whether or not the parity of the cardinality of its
connected component in D is different from that in A. Under our hypothesis,
this parity does not change, for, by (4) of Lemma 6.3.1, the type of α in C is H
or S, and so the cardinality of the connected component of α in C is odd.
(2) Generally (meaning without the assumption that O-depthC(α) is odd),
the only element of C′ that could possibly have a different type in D is the last
one of C′; it changes type if and only if it is connected to α and the cardinality
of its connected component in C′ is odd. Under our hypothesis, this cardinality
is even, for the same reason as in (1).
(3) That SD = SC′ ∪ SA (disjoint union) is an immediate consequence
of (1) and (2). By Lemma 6.3.1 (1), qA,α = qD,α dominates every element of
SA, so SA ⊆ (SD)
j0 (depth
qD,α = O-depthD(α) = O-depthC(α) = j0).
It is enough to prove the following claim: every element of SC′ has depth less
than j0 in SD. Let γ
′ be an element of SC′ . If γ
′ > qD,α then the claim is
clear. If not, then, by Proposition 5.3.4 (1), γ′ = ph(γ). By Lemma 6.3.1 (3),
O-depthD(γ) is odd. Since the claim is already true for qD,γ = pv(γ), we
have O-depthD(γ) = depthSDpv(γ) ≤ j0 − 2. By (6) of the same lemma,
depth
γ′ = O-depthD(γ)+1, so depthDγ
′ ≤ j0−1, and the claim is proved.�
Proposition 6.3.6 Let S be a monomial in ON and j an odd integer. For β
in Sj,j+1(:= {α ∈ S |O-depth
(α) ≥ j}), we have
O-depth
Sj,j+1
(β) = O-depth
(β)− j + 1
Proof: Proceed by induction on j. For j = 1, the assertion reduces to a
tautology. Suppose that the assertion has been proved upto j. By the induction
hypothesis, we have Sj+2,j+3 = (Sj,j+1)3,4, and we are reduced to proving the
assertion for j = 3.
Let A be a v-chain in S3,4 with tail β and O-depthA(β) = O-depthS3,4(β).
Let α be the head of A. We may assume that O-depth
(α) = 3 for, if
O-depth
(α) > 3, we can find, by Lemma 6.3.1 (5), α′ of O-depth 3 in S
with α′ > α, and extending A by α′ will not decrease the O-depth in A of β
(Proposition 6.1.1 (1)). Let E be a v-chain in SA with tail qA,β and length
O-depthA(β). The head of E is then qA,α (see Proposition 5.3.4 (1)).
Choose C in S with tail α such that O-depthC(α) = 3. Let D be the
concatenation of C with A. By Corollary 6.3.5, E is contained in SD, qD,α =
qA,α, and qD,β = qA,β. By Proposition 6.1.1 (2), the O-depth of α is the same
in D as in C. Choose a v-chain F in SD with tail qD,α = qA,α. Concatenating F
with E we get a v-chain inSD with tail qD,β = qA,β of lengthO-depthS3,4(β)+2.
This proves that O-depth
(β) ≥ O-depth
(β) + 2.
To prove the reverse inequality, we need only turn the above proof on
its head. Let D be a v-chain in S with tail β such that O-depth
(β) =
O-depthD(β). Let G be a v-chain in SD with tail qD,β and length O-depthS(β).
There exists an element α in D of O-depth 3 in D (by Lemma 6.3.1 (5)). Let
C be the part of D upto and including α, and A the part α > . . . > β. By
Proposition 6.1.1 (2), O-depthC(α) = 3 and, as above, Corollary 6.3.5 applies.
By Corollary 6.1.4, qA,α = qD,α occurs in G. The part F of G upto and in-
cluding qA,α is of length at most 3, and the part E : qD,α > . . . > qD,β belongs
also to SA (Proposition 5.3.4 (2)). Thus the length of G is at most 2 more than
the the length of E which is at most O-depth
(β). �
Corollary 6.3.7 For odd integers i, j, we have (Si,i+1)j,j+1 = Si+j−1,i+j . �
Corollary 6.3.8 Let E : α > . . . > ζ be a v-chain, D and D′ two v-chains with
tail α, and C, C′ the concatenations of D, D′ respectively with E. Then
1. O-depthC(ζ)−O-depthC(α) ≤ O-depthC′(ζ) −O-depthC′(α) + 1;
2. equality holds if and only if the type of λ is H in C and V in C′, and
ph(λ) > µ, where λ is the last element in the connected component con-
taining α of E and µ is the immediate successor in E of λ.
Proof: These assertions follow from combining (2) of Proposition 5.3.3 with
(1) of Lemma 6.3.1. �
Corollary 6.3.9 Let ζ be an element of a monomial S in ON. Let C be a
v-chain in S with tail ζ such that O-depthC(ζ) = O-depthS(ζ). Then
1. O-depthC(α) ≥ O-depthS(α) − 1 for any α in C.
2. If O-depthC(α) = O-depthS(α) − 1 for some α in C, then
(a) letting λ be the last element in the connected component containing α
and µ the element next to λ, the type of λ in C is H and ph(λ) > µ.
(b) O-depthC(γ) = O-depthS(γ) − 1 for all γ in C between α and λ
(both inclusive).
Proof: (1) Let α be in C. Let E denote the part of C beyond (and including) α.
Let D′ be a v-chain in S with tail α such that O-depthD′(α) = O-depthS(α).
Let C′ be the concatenation of D′ and E. Applying Proposition 6.3.8 (1), we
O-depthC(α) ≥ O-depthC(ζ) −O-depthC′(ζ) +O-depthC′(α)− 1.
But O-depthC(ζ)−O-depthC′(ζ) = O-depthS(ζ)−O-depthC′(ζ) ≥ 0, and, by
the choice of D′ and Proposition 6.1.1 (2), O-depthC′(α) = O-depthD′(α) =
O-depth
(2) Assertions (a) and (b) follow respectively from the “only if ” and “if”
parts of item (2) of Proposition 6.3.8. �
7 The map Oπ
The purpose of this section is to describe the map Oπ. The description is given
in §7.1. It relies on certain claims which are proved in §§7.3, 7.4. Those proofs
in turn refer to results from §9, but there is no circularity—to postpone the
definition of Oπ until all the results needed for it have been proved would hurt
rather than help readability. The observations in §7.5 are required only in §10.
The symbol j will be reserved for an odd positive integer throughout this
section.
7.1 Description of Oπ
The map Oπ takes as input a monomial S in ON and produces as output
a pair (w,S′), where w is an element of I(d) such that w ≥ v and S′ is a
“smaller” monomial, possibly empty, in ON. If the input S is empty, no output
is produced (by definition). So now suppose that S is non-empty.
We first partition S into subsets according to the O-depths of its elements.
Let S
be the sub-monomial of S consisting of those elements of S that have
O-depth k—the superscript “pr” is short for “preliminary”. It follows from
Corollary 6.1.3 (4) that there are no comparable elements in S
and so we can
arrange the elements of S
in ascending order of both row and column indices.
Let σk be the last element of S
in this arrangement.
Let now j be an odd integer. We set
j,j+1 := S
We say that S is truly orthogonal at j if ph(σj) belongs to N (that is, if r > r
where σj = (r, c)),
Let Sj,j+1 denote the monomial in N defined by Sj,j+1 :=
j,j+1 \ {σj}
j,j+1 \ {σj}
∪ {pv(σj), ph(σj)}
if S is truly
orthogonal at j
j,j+1 ∪
j,j+1
otherwise
Here S
j,j+1 \ {σj} and other terms on the right are to be understood as mul-
tisets. As proved in Corollary 7.3.4 (1) below, Sj,j+1 has depth at most 2.
Let Sj (respectively Sj+1 be the subset (as a multiset) of elements of depth 1
(respectively 2) of Sj,j+1.
Now, for every integer k, we apply the map of π of [7, §4] to Sk to obtain a
pair (w(k),S′k), where w(k) is an element of I(d, 2d) andS
k is a monomial in N.
Let Sw(k) be the distinguished monomial in N associated to w(k)—see §5.1.2.
Proposition 7.1.1 1. Sw(k) and S
k are symmetric. And therefore so are
∪kSw(k) and ∪kS
2. ∪kSw(k) is a distinguished subset of N (in particular, the Sw(k) are dis-
joint).
3. For j an odd integer, either
• both Sw(j) and Sw(j+1) meet the diagonal, or
• neither of them meets the diagonal,
precisely as whether or not S is truly orthogonal at j. And therefore
∪kSw(k) has evenly many diagonal elements.
4. No S′k intersects the diagonal. And therefore neither does ∪kS
The proposition will be proved below in §7.4.
Finally we are ready to define the image (w,S′) of S under Oπ. We let w
be the element of I(d, 2d) associated to the distinguished subset ∪kSw(k) of N;
since ∪kSw(k) is symmetric and has evenly many diagonal elements, it follows
from Proposition 5.2.1 that w is in fact an element of I(d). And we take S′ :=
k ∩ON.
Remark 7.1.2 Setting
π(Sj,j+1) := (wj,j+1,S
j,j+1), S
′ := ∪j oddS
j,j+1 ∩ON,
and defining w to be the element of I(d, 2d) associated to ∪j oddSwj,j+1 would give
an equivalent definition of Oπ.
7.2 Illustration by an example
We illustrate the map Oπ by means of an example. Let d = 15, and v =
(1, 2, 3, 4, 9, 10, 14, 16, 18, 19, 20, 23, 24, 25, 26). A monomial S in ON is shown
in Figure 7.2.1. Solid black dots indicate the elements that occur in S with non-
zero multiplicity. Integers written near the solid dots indicate multiplicities.
The O-depth of S is 5. The element (21, 9) has O-depth 3 although it has
depth 2 in S. Figure 7.2.2 shows the monomials S
1,2, S
3,4, and S
5,6. Solid
dots, open dots, and crosses indicate elements of these monomials respectively.
The monomial S is truly orthogonal at 1 and 3 but not at 5: σ1 = (28, 2),
σ3 = (21, 9), and σ5 = (15, 14).
Figure 7.2.3 shows the monomials S1,2, S3,4, and S5,6 of N and also their
decomposition into blocks, and Figure 7.2.4 the monomialsS′1,2, S
3,4, and S
We have
Sw = {(15, 14), (17, 16), (21, 10), (7, 4), (27, 24), (28, 3), (30, 1), (29, 2)}
hence w = (7, 9, 15, 17, 18, 19, 20, 21, 23, 25, 26, 27, 28, 29, 30). It is easy to check
that w ∈ I(d). The monomial S′ is the intersection with ON of the union of
S′1,2, S
3,4, and S
5,6—in other words it is just the monomial lying above d in
Figure 7.2.4.
diagonal
1         2        3        4       9       10      14     16     18      19     20     23     24      25     26  
2 1 3 4
3 2 6 2
2 3 2
3 1 2 3
Figure 7.2.1: The monomial S
diagonal
1         2        3        4       9       10      14     16     18      19     20     23     24      25     26  
2 1 3 4
3 2 6 2
2 1 3 2
3 1 2 3
Figure 7.2.2: S
1,2, S
3,4, and S
diagonal
1         2        3        4       9       10      14     16     18      19     20     23     24      25     26  
2 1 3 4
3 2 6 2
2 1 3 2
3 1 2
2 4 3 1 4 2 1 3 2
1 2 1 3 2 1
2 4 6 5 3
1 2 1 4
Figure 7.2.3: S1,2, S3,4, and S5,6
diagonal
1         2        3        4       9       10      14     16     18      19     20     23     24      25     26  
2 1 4
3 6 5 3
2 1 3 2 1
1 4 3 1 4 2 1 3 2
Figure 7.2.4: S′1,2, S
3,4, and S
7.3 A proposition about Sj,j+1
The aim of this subsection is to show that Sj,j+1 has depth no more than 2—
see item (1b) of Proposition 7.3.3. This basic fact was mentioned above in
the description of Oπ and is necessary (psychologically although not logically)
to make sense of the definitions of Sj and Sj+1. We prepare the way for
Proposition 7.3.3 by way of two preliminary propositions. The first of these is
about elements of O-depth j and j + 1 in S, the second about the relation of
these elements with σj .
Proposition 7.3.1 1. S
has no comparable elements.
2. For j an odd integer and β an element of S
j+1, there exists α in S
j such
that α > β. In particular, the row index of σj+1 (if σj+1 exists) is less
than the row index of σj.
Proof: (1) follows from Corollary 6.1.3 (4); (2) follows from Proposition 6.3.3
and Lemma 6.3.1 (5). �
Proposition 7.3.2 Let j be an odd integer and let S be truly orthogonal at j.
1. pv(σj) > ph(σj); if α > pv(σj), then α > σj; if α > σj , then α > ph(σj).
2. No element of S
j is comparable to pv(σj) or ph(σj).
3. No element of S
j+1 is comparable to ph(σj).
4. The following is not possible: α ∈ S
j , β ∈ S
j+1, and ph(α) > β.
Proof: (1) is trivial. (2) follows immediately from the definition of σj . We
now prove (3). First suppose β > ph(σj) for some β in S
j+1. By (2) of
Proposition 7.3.1, there exists α in S
j such that α > β. But then the row
index of α exceeds that of σj , a contradiction to the choice of σj .
We claim that it is not possible for β ∈ S
j+1 to satisfy ph(σj) > β. This
being a special case of (4), we need only prove that statement. So suppose that
α belongs to S
j and that ph(α) > β. Let C be a v-chain in S with tail α such
that O-depthC(α) = j (see Proposition 6.1.3 (1)). Concatenate C with α > β
and call the resulting v-chain D. Then, by Lemma 6.3.1 (4), α is of type H
in D, so that, by Lemma 6.3.1 (1), we have O-depthD(β) = O-depthD(α) + 2.
But, by Proposition 6.1.1 (2), O-depthD(α) = O-depthC(α) = j, so that
O-depth
(β) ≥ j + 2, a contradiction. �
Let Sj,j+1(ext) denote the set—not multiset—defined by:
Sj,j+1(ext) :=
Sj,j+1 ∪ {σj , σ
j } if S is truly orthogonal at j
Sj,j+1 otherwise
Here Sj,j+1 on the right stands for the underlying set of the multiset Sj,j+1
defined above. The set Sj,j+1(ext) is the disjoint union of the sets Sj(ext)
and Sj+1(ext) defined as follows (here again the terms on the right hand side
denote the underlying sets of the corresponding multisets):
Sj(ext) :=
∪ {pv(σ)} if S is truly orthogonal at j
otherwise
Sj+1(ext) :=
j+1 ∪
∪ {ph(σ)} if S is truly orthogonal at j
j+1 ∪
otherwise
Proposition 7.3.3 1. Sj(ext) (respectively Sj+1(ext)) is precisely the set
of elements of depth 1 (respectively 2) in Sj,j+1(ext). In particular,
(a) Neither Sj(ext) nor Sj+1(ext) contains comparable elements.
(b) The length of a v-chain in Sj,j+1(ext) is at most 2.
(c) There is a v-chain of length 2 in Sj,j+1 unless Sj+1(ext) is empty.
2. Let k be a positive integer, not necessarily odd. If there is in S an element
of O-depth at least k, then Sk(ext) is non-empty. The converse also holds
except possibly if k is even and S is truly orthogonal at k−1. In particular,
if Sk(ext) is non-empty, then there is an element of O-depth at least k−1.
Proof: (1): It is enough to show that every element of Sj(ext)(up) (respec-
tively Sj+1(ext)(up)) is of depth 1 (respectively 2) in Sj,j+1(ext)(up), for
• α > β implies α(up) > β(up) for elements α, β of N.
• Sj,j+1(ext) = Sj(ext) ∪Sj+1(ext).
• Sj,j+1(ext), Sj(ext), and Sj+1(ext) are symmetric.
In turn, it is enough to show the following:
(i) Every element of Sj(ext)(up) has depth 1.
(ii) Sj+1(ext)(up) has no comparable elements.
(iii) Every element of Sj+1(ext)(up) has depth at least 2.
Item (i) follows from Proposition 7.3.1 and Proposition 7.3.2 (2); item (ii)
from Proposition 7.3.1 (1) and Proposition 7.3.2 (3); item (iii) from Propo-
sition 7.3.1 (2) and Proposition 7.3.2 (1).
(2): The first assertion follows from Lemma 6.3.1 (5): if k is odd there is an
element ofO-depth k inS; if k is even and there is no element ofO-depth k inS,
then there is in S an element of O-depth k− 1 and of type H, so S is truly or-
thogonal at k−1. The second assertion is clear from the definition of Sk(ext). �
Corollary 7.3.4 1. No element of Sj,j+1 has depth more than 2.
2. Sj+1(ext) = Sj+1 and Sj(ext) ∩ Sj,j+1 = Sj (as sets). In particular,
Sj+1 = Sj,j+1∩Sj+1(ext) and Sj = Sj,j+1∩Sj(ext) as multisets defined
by the intersection of a multiset with a subset.
Proof: (1): Since Sj,j+1 ⊆ Sj,j+1(ext) (as sets), this follows immediately
from (1b) of the proposition above.
(2): Since the union of Sj+1(ext) (which always is contained in Sj,j+1) and
Sj(ext) ∩Sj,j+1 is all of Sj,j+1, and since Sj , Sj+1 are disjoint, it it enough
to show that Sj+1(ext) ⊆ Sj+1 and Sj(ext) ∩Sj,j+1 ⊆ Sj .
Now, since elements of Sj(ext) have depth 1 even in Sj,j+1(ext) (by item (1)
of the proposition above), it is immediate that Sj(ext) ∩ Sj,j+1 ⊆ Sj . And
it follows from the proof of item (iii) in the proof of item (1) of the proposi-
tion above that an element of Sj+1(ext) has depth 2 even in Sj,j+1 (not just
in Sj,j+1(ext)), so that Sj+1(ext) ⊆ Sj+1. �
7.4 Proof of Proposition 7.1.1
(1) The monomials Sj,j+1 are clearly symmetric. Observe that α in Sj,j+1 has
the same depth as α#, for α1 > α2 implies α(up) > α2(up) and α(down) >
α2(down) for α1, α2 in N. Thus the monomials Sk are symmetric. Since the
map π of [7] respects #—see Proposition 5.7 of [4]—it follows that Sw(k) and
S′k are symmetric. Therefore so are ∪kSw(k) and ∪kS
(2) This follows from Corollary 9.3.6.
(3) IfS is truly orthogonal at j, then pv(σj) and ph(σj) are diagonal elements
respectively in Sj and Sj+1—see Corollary 7.3.4 (2). Thus both Sj and Sj+1
have diagonal blocks in the sense of Proposition 5.10 (A) of [4]. It follows from
the result just quoted that both Sw(j) and Sw(j+1) meet the diagonal. It is
of course clear that each Sw(k) meets the diagonal at most once since diagonal
elements are clearly comparable but elements of Sw(k) are not by Lemma 4.9
of [7].
Suppose that S is not truly orthogonal at j. Then σj and σ
j belong to
different blocks—this is equivalent to the definition of S being not truly orthog-
onal at j. By Proposition 7.3.1 (2), it follows that σj+1 and σ
j+1 also belong
to different blocks. So neither Sj nor Sj+1 has a diagonal block.
(4) If S is not truly orthogonal at j, then neither Sj nor Sj+1 has a diagonal
block (as has just been said above), and it follows from Proposition 5.10 (A)
of [4] that neither S′j nor S
j+1 meets the diagonal.
So suppose that S is truly orthogonal at j. Then both Sj and Sj+1 have
a diagonal entry each of multiplicity 1, namely pv(σj) and ph(σj) respectively.
It is clear from the definition of σj that no element of Sj(up) shares its row
index with pv(σj). And it follows from Proposition 7.3.1 (2) that no element
of Sj+1(up) shares its row index with ph(σj). It now follows from the proof of
Proposition 5.10 (B) of [4]—see the last line of that proof—that neither S′j nor
S′j+1 meets the diagonal. �
7.5 More observations
Proposition 7.5.1 The length of any v-chain in Sj,j+1∪S
j+1 is at most 2.
Proof: By Corollary 7.3.4 (1), the length of any v-chain in Sj,j+1 is at most 2.
Applying Lemma 9.1.1 to Sj,j+1, we get the desired result. �
Proposition 7.5.2 1. For an element α′ = (r, c) of S′k(up), there exists an
element α = (r, C) of S
with C ≤ c.
2. For an element α′ = (r, c) of S′j+1(up), there exists an element α = (R, c)
of Sj+1(up) with r ≤ R.
3. For an element α′ of S′j+1(up), there exists an element α of S
j with
α > α′.
Proof: (1) That there exists α inSk(up) with C ≤ c follows from the definition
of S′k(up). Clearly such an α cannot be on the diagonal, so α belongs to S
(2) As in the proof of (1), it follows from the definition of S′j+1 that there
exists α = (R, c) in Sj+1 with r ≤ R. If α lies strictly below the diagonal, then
c > R∗, so that α∗ = (c∗, R∗) > α′ = (r, c), a contradiction to Lemma 9.1.1 (α∗
belongs to Sj+1 by the symmetry of Sj+1). Thus α belongs to Sj+1(up).
(3) Writing α′ = (r, c), by (1), we can find an β = (r, C) in S
j+1 with C ≤ c.
By Proposition 7.3.1 (2), there exists α in S
j,j+1 such that α > β. �
Corollary 7.5.3 If in S′j(up)∪S
j+1(up) there exists an element with horizon-
tal projection in N, then S is truly orthogonal at j.
Proof: Follows directly from Proposition 7.5.2 (1) and (3). �
Proposition 7.5.4 The O-depth of an element in S
j ∪ S
j+1 is at most 2.
More strongly, the O-depth of an element in S
j,j+1 ∪S
j(up) ∪S
j+1(up) is at
most 2.
Proof: It is enough to show that no element in S′j(up)∪S
j+1(up) hasO-depth
more than 2, for we may assume by increasing multiplicities that S
j ⊆ S
j(up)
and S
j+1 ⊆ S
j+1(up) (as sets). It follows from Proposition 7.5.1 that a v-chain
in S′j(up) ∪ S
j+1(up) has length at most 2. Let α
1 = (r1, c1) > α
2 = (r2, c2)
be such a v-chain. It follows from the proof of Corollary 4.14 (2) of [7] that
α′1 ∈ S
j(up) and α
2 ∈ S
j+1(up). By item (1) of Lemma 6.3.1, it is enough to
rule out the following possibility: α′1 is of type H in α
1 > α
2 and ph(α
1) > α
Suppose that this is the case. By Proposition 7.5.2 (1) and (2), it follows
that there exist elements α1 = (r1, C1) ∈ S
j,j+1 and α2 = (R2, c2) ∈ Sj+1(up)
with C1 ≤ c1 and r2 ≤ R2. Since ph(α
1) > α
2, it follows that α1 > α2. Now, if
α2 = ph(σj), then Proposition 7.3.2 (2) is contradicted; if α2 belongs to S
Proposition 7.3.2 (4) is contradicted (because ph(α1) > α2). �
8 The map Oφ
The purpose of this section is to describe the map Oφ and prove some basic
facts about it. Certain proofs here refer to results from §9, but there is no
circularity—to postpone the definition of Oφ until all the results needed for it
have been proved would hurt rather than help readability. As in §7, the symbol j
will be reserved for an odd integer throughout this section.
8.1 Description of Oφ
The map Oφ takes as input a pair (w,T), where T is a monomial, possibly
empty, in ON and w ≥ v an element of I(d) that O-dominates T, and produces
as output a monomial T∗ of ON. To describe Oφ, we first partition T into
subsets Tw,j,j+1. As the subscript w in Tw,j,j+1 suggests, this partition depends
on w.
For an odd integer j, let Sjw (respectively Sw,j,j+1) denote the subset of Sw
consisting of those elements that are j-deep (respectively that are j deep but
not j + 2 deep, or equivalently of depth j or j + 1) in Sw in the sense of [7,
§4]. Since Sw is distinguished, symmetric, and has evenly many elements on
the diagonal d, it follows that Sjw and Sw,j,j+1 too have these properties, and
that, in fact, the number of diagonal elements of Sw,j,j+1 is either 0 or 2 (in
the latter case, the elements have to be distinct since Sw is distinguished and
so is multiplicity free). Let us denote by wj and wj,j+1 the elements of I(d)
corresponding to Sjw and Sw,j,j+1 by Proposition 5.2.1.
Let Tw,j,j+1 denote the subset of T consisting of those elements α such that
• every v-chain in T with head α is O-dominated by wj , and
• there exists a v-chain in T with head α that is not O-dominated by wj+2.
It is evident that the subsets Tw,j,j+1 are disjoint (as j varies over the odd
integers) and that their union is all of T (for w = w1 O-dominates all v-chains
in T by hypothesis and Sjw is empty for large j and so w
j = v). In other words,
the Tw,j,j+1 form a partition of T.
Lemma 8.1.1 1. The length of a v-chain in Tw,j,j+1∪T
w,j,j+1 is at most 2.
In fact, the O-depth of any element in Tw,j,j+1 is at most 2.
2. wj,j+1 O-dominates Tw,j,j+1.
Proof: The lemma follows rather easily from Corollary 9.2.3 as we now show.
Let C be a v-chain in Tw,j,j+1. Let τ be the tail of C. Choose a v-chain D
in T with head τ that is not O-dominated by wj+2. Let E be the concate-
nation of C with D. Since the head of E belongs to Tw,j,j+1, it follows that
E is O-dominated by wj . It follows from (the only if part of) Corollary 9.2.3
(applied with S = E and x = wj) that wj,j+1 O-dominates E
1 ∪ E
2 and
wj+2 O-dominates E3,pr. This means τ 6∈ E3,pr, so τ ∈ E
1 ∪ E
2 , and so
C ⊆ E
1 ∪ E
2 . This proves (2). By Proposition 6.1.1 (2), the O-depths of
elements of C are the same in C and E, so C ⊆ C
1 ∪ C
2 , which proves the
second assertion of (1). The first assertion of (1) follows from the second (see
Lemma 6.2.1). �
Corollary 8.1.2 wj,j+1 dominates Tw,j,j+1 ∪ T
w,j,j+1 in the sense of [7].
Proof: This follows from (2) of Lemma 8.1.1 and Corollary 5.3.6 (the latter
applied with S = Tw,j,j+1 and w = wj,j+1). �
We may therefore apply the map φ of [7, §4] to the pair (wj,j+1,Tw,j,j+1 ∪
w,j,j+1) to obtain a monomial (Tw,j,j+1∪T
w,j,j+1)
⋆ in N. In applying φ, there
is the partitioning of Tw,j,j+1 ∪ T
w,j,j+1 into “pieces”, these being indexed by
elements of Swj,j+1 = Sw,j,j+1—observe that the elements of depth 1 (respec-
tively 2) of Sw,j,j+1 are precisely those of Sw of depth j (respectively j + 1).
We denote by Pβ the piece of Tw,j,j+1∪T
w,j,j+1 corresponding to β in Swj,j+1 .
We also use the notation P∗β as in [7]. Moreover, we will use the phrase piece
of T (with respect to w being implicitly understood) to refer to a piece of
Tw,j,j+1 ∪ T
w,j,j+1 for some odd integer j.
Caution: Thinking of T as a monomial in N and w as an element of I(d, 2d) that
dominates it, there is, as in [7], the notion of “piece of T” (with respect to w).
The two notions of “piece” are different.
Lemma 8.1.3 1. The monomial (Tw,j,j+1∪T
w,j,j+1)
⋆ is symmetric and has
either none or two distinct diagonal elements depending exactly on whether
Swj,j+1 = Sw,j,j+1 has 0 or 2 elements on the diagonal.
2. The depth of (Tw,j,j+1 ∪ T
w,j,j+1)
⋆ is 2; and ∪β∈(Sw)jP
β, ∪β∈(Sw)j+1P
are respectively the elements of depth 1 and 2 in (Tw,j,j+1 ∪ T
w,j,j+1)
Proof: (1) The symmetry follows by combining Proposition 5.6 of [4], which
says that the map π respects the involution #, with Proposition 4.2 of [7], which
says that π and φ are are inverses of each other.
The assertion about diagonal elements follows by combining item (B) of [4,
Proposition 5.10], which is an assertion about the existence and relative mul-
tiplicities of diagonal elements in B and B′ where B is a diagonal block of a
monomial in N, and Proposition 4.2 of [7].
(2) It follows from Propositions 4.2 of [7] that the map π (described in §4 of
that paper) applied to (Tw,j,j+1∪T
w,j,j+1)
⋆ results in the pair (wj,j+1,Tw,j,j+1∪
Tw,j,j+1). It now follows from Lemma 4.16 of [7] that the depth of (Tw,j,j+1 ∪
w,j,j+1)
⋆ is exactly 2. The latter assertions again follow from the results of [7]—
in fact, the proof that π ◦ φ is identity on pages 47–49 of [7] shows that the P∗β
are the blocks in the sense of [7] of the monomial (Tw,j,j+1 ∪ T
w,j,j+1)
Suppose that (Tw,j,j+1 ∪ T
w,j,j+1)
⋆ contains the pair (a, a∗), (b, b∗) of di-
agonal elements with a > b. We call the pair (b, a∗), (a, b∗) the “twists,” and
set δj := (b, a
∗). In other words, δj is the element of the twisted pair that lies
above the diagonal—observe that the twisted elements are reflections of each
other. We allow ourselves the following ways of expressing the condition that
(Tw,j,j+1 ∪ T
w,j,j+1)
⋆ has diagonal elements: δj exists; w is diagonal at j (the
latter expression is justified by the lemma above).
With notation as above, consider the new monomial defined as
(Tw,j,j+1 ∪ T
w,j,j+1)
⋆ if w is not diagonal at j(
(Tw,j,j+1 ∪ T
w,j,j+1)
⋆ \ d
∪ {δj, δ
j } if w is diagonal at j
This new monomial is symmetric and contains no diagonal elements. Its inter-
section with ON is denoted T⋆w,j,j+1. In other words, T
w,j,j+1 is the intersection
of the new monomial with the subset of N of those elements that lie strictly
above the diagonal.
The union of T⋆w,j,j+1 over all odd integers j is defined to be T
w, the result
of Oφ applied to (w,T). This finishes the description of the map Oφ.
For β in Sw,j,j+1(up), we define the “orthogonal piece-star” OP
β corre-
sponding to β as
OP∗β :=
P∗β = P
β(up) if β is not on the diagonal
P∗β ∩ON if β ∈ (Sw)j+1 is on the diagonal
{P∗β ∩ON} ∪ {δj} if β ∈ (Sw)j is on the diagonal
(8.1.1)
With this, we can say that T⋆w is the union of OP
β as β varies over Sw(up).
Lemma 8.1.4 Suppose that (Tw,j,j+1 ∪ T
w,j,j+1)
⋆ contains the pair (a, a∗),
(b, b∗) of diagonal elements with a > b. Let
. . . , (r1, c1), (a, a
∗), (c∗1, r
1), . . . ; . . . , (r2, c2), (b, b
∗), (c∗2, r
2), . . .
be respectively the elements of depth 1 and 2 of (Tw,j,j+1 ∪ T
w,j,j+1)
⋆ arranged
in increasing order of row and column indices. Then
1. c1 ≤ a
∗ and r1 ≤ b (assuming (r1, c1) exists); and
2. r2 < b and c2 ≤ b
∗ (assuming (r2, c2) exists).
Proof: (1) Suppose that (r1, c1) exists. It is clear that c1 ≤ a
∗. From way the
map φ of [7] is defined, it follows that (r1, a
∗) is an element of Tw,j,j+1. Suppose
that r1 > b. Then ph(r1, a
∗) = (r1, r
1) belongs to N. We consider two cases.
If (r2, c2) exists, then, again from the definition of the map φ, it follows that
(r2, b
∗) is an element of Tw,j,j+1. But then ph(r1, a
∗) = (r1, r
1) > (b, b
∗) and
(b, b∗) dominates (r2, b
∗), which means that the v-chain (r1, a
∗) > (r2, b
∗) (note
that a∗ < b∗ because a > b by hypothesis) in Tw,j,j+1 has O-depth more than 2,
a contradiction to Lemma 8.1.1 (1).
Now suppose that (r2, c2) does not exist. (Then (b, b
∗) is the diagonal ele-
ment in (Sw)j+1.) Consider the singleton v-chain C := {(r1, a
∗)} in Tw,j,j+1.
Then SC = {(a, a
∗), (r1, r
1)} which is not dominated by wj,j+1, a contradiction
to Lemma 8.1.1 (2).
(2) Suppose that (r2, c2) exists. Then there exists, by the definition of the
map φ, an element (r2, b
∗) in Tw,j,j+1. Since (r2, b
∗) lies above the diagonal, it
follows that r2 < b. That c2 ≤ b
∗ is clear. �
8.2 Basic facts about Tw,j,j+1 and T
w,j,j+1
Lemma 8.2.1 1. Let α′ > α be elements of T. Let j and j′ be the odd
integers such that α′ ∈ Tw,j′,j′+1 and α ∈ Tw,j,j+1. Then j
′ ≤ j.
2. If, further, either
(a) there exists µ in T such that α′ > µ > α, or
(b) α′ ∈ Pβ′ for β
′ in (Sw)j′+1,
then j′ < j.
Proof: (1) By hypothesis, every v-chain with head α′ is O-dominated by wj
This implies, by Corollary 6.1.2, that every v-chain with head α is O-dominated
by wj
. This shows j′ ≤ j.
(2a) Suppose that j′ = j. It follows from (1) that α′, µ, and α all belong
to Tw,j,j+1. But then α
′ > µ > α is a v-chain of length 3 in Tw,j,j+1, a
contradiction to Lemma 8.1.1 (1).
(2b) Suppose that j′ = j. Then α′ > α is a v-chain in Tw,j,j+1. Being of
length 2, it cannot be dominated by (Sw)j+1, which means, by the definition
of Pβ′ , that α
′ cannot belong to Pβ′ , a contradiction. �
Proposition 8.2.2 1. The length of a v-chain in T⋆w,j,j+1 is at most 2.
2. The O-depth of T⋆w,j,j+1 is at most 2.
3. ∪β∈(Sw)j(up)OP
β is precisely the set of depth 1 elements of T
w,j,j+1 (in
particular, no two elements there are comparable); if δj exists, then it is
the last element of ∪β∈(Sw)j(up)OP
β when the elements are arranged in
increasing order of row and column indices.
4. ∪β∈(Sw)j+1(up)OP
β is precisely the set of depth 2 elements of T
w,j,j+1 (in
particular, no two elements there are comparable); if δj exists, then its
row index exceeds the row index of any element in ∪β∈(Sw)j+1(up)OP
Proof: For (1), it is enough, given Lemma 8.1.3 (2), to show that δj is not
comparable to any element of depth 1 of (Tw,j,j+1 ∪T
w,j,j+1)
⋆, and this follows
from Lemma 8.1.4 (1). In fact, the above argument proves also (3).
For (4), it is enough, given Lemma 8.1.3 (2), the symmetry of the monomi-
als involved in that lemma, and the observation that α > β implies α(up) >
β(up) for elements α, β of N, to show the following: if (a, a∗) > γ = (e, f)
for γ an element of (Tw,j,j+1 ∪ T
w,j,j+1)
⋆ lying (strictly) above the diagonal,
then δj > γ. But this follows from Lemma 8.1.4 (2): γ is a depth 2 ele-
ment in (Tw,j,j+1 ∪ T
w,j,j+1)
⋆, and we have e ≤ r2 < b (and a
∗ < f since
(a, a∗) > γ). In fact, the above argument proves also (2): observe that f ≤ b∗
(Lemma 8.1.4 (2)). �
9 Some Lemmas
The main combinatorial results of this paper are Propositions 4.1.1 and 4.1.2.
They are analogues respectively of Propositions 4.1 and 4.2 of [7]. We have
tried to preserve the structure of the proofs in [7] of those propositions. The
proofs in [7] rely on certain lemmas and it is natural therefore to first establish
the orthogonal analogues of those. The purpose of this section is precisely that.
Needless to say that the lemmas (especially those in §9.4) may be unintelligible
until one tries to read §10.
The division of this section into four subsections is also suggested by the
structure of the proofs in [7]. Each subsection has at its beginning a brief
description of its contents.
9.1 Lemmas from the Grassmannian case
In this subsection, the terminology and notation of [7, §4] are in force. The state-
ments here could have been made in [7, §4] and would perhaps have improved
the efficiency of the proofs there, but do not appear there explicitly.
Let S be a monomial in N. Recall from [7] the notion of depth of an ele-
ment α in S: it is the largest possible length of a v-chain in S with tail α and
denoted depth
α. The depth of S is the maximum of the depths in it of all its
elements. We denote by Sk the set of elements of depth k of S (as in [7]) and
by Sk the set of elements of depth at least k of S.
Caution: For a monomial S of ON, we have introduced in §7.1 the notation Sk.
That is different from the Sk we have just defined.
Lemma 9.1.1 Let S be a monomial in N, and let π(S) = (w,S′), where π is
the map defined in [7, §4]. Then the maximum length of a v-chain in S∪S′ is
the same as the maximum length of a v-chain in S.
Proof: We use the notation of [7, §4] freely. Let d be the maximum length of
a v-chain in S. Suppose α1 > . . . > αℓ is a v-chain in S ∪S
′. Let i1, . . . , iℓ be
such that αj belongs to Sij ∪ S
(the integers ij are uniquely determined—
see Corollary 5.4 of [4]). We claim that i1 < . . . < iℓ. This suffices to prove the
lemma, for Sk ∪S
k is empty for k > d.
To prove the claim, it is enough to show i1 < i2. It follows from Lemma 4.10
of [7] that i1 6= i2. We now assume that i1 > i2 and arrive at a contradiction.
First suppose that α1 ∈ Si1 . Then, by the definition ofSi1 , there exists β inSi2
with β > α1. Now β > α2 and both β, α2 belong to Si2 ∪S
, a contradiction
to [7, Lemma 4.10]. If α1 = (r, c) belongs to S
, then, by the definition of S′i1 ,
there exists (r, a) in Si1 with a ≤ c, and there exists β in Si2 with β > (r, a).
This leads to the same contradiction as before. �
Lemma 9.1.2 Let B and U be monomials in N. Assume that
• the elements of B form a single block (in the sense of [7, Page 38]).
• U has depth 1 (equivalently, there are no comparable elements in U).
• for every β = (r, c) in B, there exist γ1(β) = (R1, C1), and γ2(β) =
(R2, C2) in U such that
C1 < c, C2 < R1, r < R2
(this holds, for example, when there exists γ(β) in U such that γ(β) > β:
take γ1(β) = γ2(β) = γ(β)).
Then there exists a unique block C of U such that w(C) > w(B).
Proof: It is useful to isolate the following observation:
Lemma 9.1.3 Let (r1, c1) and (r2, c2) be elements of N with c2 < r1 ≤ r2. Let
γ11 = (R
1 ), γ
1 = (R
1 ) and γ
2 = (R
2 ), γ
2 = (R
2 ) be elements
of N such that
1. C11 ≤ c1, C
1 < R
1, r1 ≤ R
2. C12 ≤ c2, C
2 < R
2, r2 ≤ R
3. No two of γ11 , γ
1 , γ
2 , γ
2 are comparable (they could well be equal and this
is important for us—see our definition of comparability).
Then the monomial {γ11 , γ
1 , γ
2 , γ
2} consists of a single block.
Proof: It follows from assumption (1) that γ11 and γ
1 belong to a single block:
• if R11 < R
1, then C
1 < R
1 becomes relevant;
• if R21 < R
1, then the other two inequalities in (1) become relevant:
C11 ≤ c1 < r1 ≤ R
Similarly it follows from assumption (2) that γ12 and γ
2 belong to a single block.
We therefore need only consider the cases when, in the arrangement of the
elements {γ11 , γ
1 , γ
2 , γ
2} in increasing order of row indices, both γ
1 , γ
1 come
before or after γ12 , γ
2 . In the former case, the first sequence of inequalities
below shows that γ21 and γ
2 belong to the same block, and we are done; in
the latter case, the second sequence of inequalities below shows that γ22 and γ
belong to the same block, and we are done:
• C12 ≤ c2 < r1 ≤ R
• C11 ≤ c1 < r1 ≤ r2 ≤ R
Continuing with the proof of Lemma 9.1.2, we first prove the existence part.
Arrange the elements of B in non-decreasing order of row numbers as well as
column numbers (this is possible since there are no comparable elements in B).
If β1 = (r1, c1) and β2 = (r2, c2) are successive elements, then c2 < r1 ≤ r2
(since B is a single block). Apply Lemma 9.1.3 with γ11 = γ
1(β1), γ
1 = γ
2(β1),
and γ12 = γ
1(β2), γ
2 = γ
1(β2). We conclude that {γ
1 , γ
1 , γ
2 , γ
2} belongs to a
single block, say C, of U. Continuing thus, we conclude that all γ1(β) and γ2(β),
as β varies over B, belong to C. Since the row (respectively column) index of
w(C) is the maximum (respectively minimum) of all row (respectively column)
indices of elements of C (and similarly for B), it follows that w(C) > w(B).
To prove uniqueness, let C1 and C2 be two blocks of U with w(C1) > w(B)
and w(C2) > w(B). Apply the lemma with (r1, c1) = (r2, c2) = w(B) and
γ11 = γ
1 = w(C1) and γ
2 = γ
2 = w(C2); it follows from [7, Lemma 4.9] that
w(C1) and w(C2) are not comparable. But, unless C1 = C2, neither is the mono-
mial {w(C1), w(C2)} a single block, again by [7, Lemma 4.9]. �
Lemma 9.1.4 Let S be a monomial in N and x an element of I(d, n). For
x to dominate S it is necessary and sufficient that for every α = (r, c) in S
there exist β = (R,C) in Sx with C ≤ c, r ≤ R, and depthSxβ ≥ depthSα.
(Here Sx denotes the distinguished monomial in N associated to x as in [7,
Proposition 4.3].)
Proof: The lemma is a corollary of [7, Lemma 4.5] as we now show.
First suppose that x dominatesS. Let α = (r, c) be an element ofS, and C a
v-chain in S with tail α and length depth
α. Since x dominates C, there exists,
by [7, Lemma 4.5], a chain in D in Sx of length depthSα and tail β = (R,C)
with C ≤ c and r ≤ R, and we are done with the proof of the necessity.
To prove the sufficiency, let C : α1 = (r1, c1) > . . . > αk = (rk, ck) be a
v-chain in S. By hypothesis, there exist β1 = (R1, C1), . . . , βk = (Rk, Ck) in
Sx with Ci ≤ ci, ri ≤ Ri, and depthSxβi = i for 1 ≤ i ≤ k (observe that
replacing the ≥ in the latter condition of the statement by an equality yields
an equivalent statement). We claim that β1 > . . . > βk. By [7, Lemma 4.5], it
suffices to prove the claim.
Since βk has depth k in Sx, there exists a β
k−1 = (R
k−1, C
k−1) of depth
k − 1 in Sx such that β
k−1 > βk. It follows from the distinguishedness of Sx
that that β′k−1 = βk−1: if not, then we have two distinct elements of the same
depth (namely k−1) in Sx both dominating αk, a contradiction. So βk−1 > βk,
and the claim is proved by continuing in a similar fashion. �
Let x be an element of I(d, n). Let Sx denote the distinguished monomial
in N associated to x as in [7, Proposition 4.3]. For k a positive integer, let xk
denote the element of I(d, n) corresponding to the distinguished subset (Sx)k.
For a monomial S ofN, let Sk,k+1 := Sk∪Sk+1. Let xk,k+1 denote the element
of I(d, n) corresponding to the distinguished monomial (Sx)k,k+1; let x
k denote
the element of I(d, n) corresponding to the distinguished subset (Sx)
Caution: For a monomial S of ON and an odd integer j, we have introduced
in §7.1 the notation Sj,j+1. That is different from the Sk,k+1 just defined.
Corollary 9.1.5 x dominates S ⇔ xk dominates Sk ∀ k ⇔ x1,2 dominates
S1,2 and x
3 dominates S3.
Proof: The first equivalence is a restatement of the lemma: in the statement
of the lemma we could equally well have written depth
β = depth
α. The
second follows from the first and the following observations: (S1,2)1 = S1,
(S1,2)2 = S2, (S
3)k = Sk+2; and (x1,2)1 = x1, (x1,2)2 = x2, (x
3)k = xk+2. �
9.2 Orthogonal analogues of Lemmas of 9.1
Lemma 9.2.2 below is the orthogonal analogue of Lemma 9.1.4 (more precisely,
that of the first assertion of Corollary 9.1.5). The following proposition will be
used in its proof.
Proposition 9.2.1 Let x be an element of I(d) and S a monomial in ON.
Then x O-dominates S
1 ∪ S
2 if and only if it O-dominates every v-chain
in S of O-depth at most 2.
Proof: The “if” part is immediate from definitions (in any case, see also
Proposition 7.5.4). For the “only if” part, let C be a v-chain in S of O-depth
at most 2. Our goal is to show that x dominates SC . For this, it is enough, by
Corollary 9.1.5, to show that x1 dominates (SC)1 and x2 dominates (SC)2 (by
choice of C, (SC)k is empty for k ≥ 3).
Let α′ ∈ (SC)1. Choose α in C such that α
′ ∈ SC,α. Choose α0 in S
1 such
that α0 dominates α. Since x O-dominates the singleton v-chain {α0}, it follows
that x1 dominates q{α0},α0 . We claim that q{α0},α0 dominates α
′. To prove the
claim, we need only rule out the possibility that α0 is of type S in {α0} and α
of type V in C. Since α′ ∈ (SC)1, it follows from Proposition 5.3.4 (1) that α
is the first element of C. In particular, if α is of type V in C, then ph(α) ∈ N,
so ph(α0) ∈ N, and α0 is of type H in {α0}. The claim is thus proved.
Now consider an element of (SC)2. Observe that the length of C is at most 2
(Lemma 6.2.1). So our element is either the horizontal projection ph(α) of the
head α of C, or it is qC,β where β is the tail of C. In the first case, let α0 be
as in the previous paragraph, and proceed similarly. It is clear that ph(α0) ∈ N
(because ph(α) ∈ N); x2 dominates ph(α0) and so also ph(α).
Now we handle the second case. If β ∈ S
2 , then C is contained in S
and there is nothing to prove. So assume that O-depth
(β) ≥ 3. Choose a
v-chain D in S with tail β, O-depthD(β) ≥ 3, and with the good property
as in Proposition 6.3.3. There occurs in D an element of O-depth 3, say δ.
(Lemma 6.3.1 (5)). Let A denote the part δ > . . . of D and C′ the part up to
but not including δ. There clearly is an element—call it µ—of depth 2 in SD
that dominates qD,β. This element µ belongs to SC′ (Corollary 6.3.5 (3)). Since
D has the good property of Proposition 6.3.3, C′ ⊆ S
2 , so µ is dominated
by an element in (Sx)2. In particular, qD,β is dominated by the same element
of (Sx)2.
We are still not done, for it is possible that qD,β be β and qC,β be pv(β).
Suppose that this is the case. Then α > β is connected. So ph(α) ∈ N and the
legs of α and β intertwine. As seen above in the third paragraph of the present
proof, there is an element of (Sx)2 that dominates ph(α). By the distinguished-
ness of Sx, it follows that the element in (Sx)2 dominating β is the same as
the one dominating ph(α). By the symmetry of Sx, this element lies on the
diagonal and so dominates pv(β), and, finally, we are done with the proof in the
second case. �
Lemma 9.2.2 Let S be a monomial in ON and x an element of I(d). For x to
O-dominate S it is necessary and sufficient that, for every odd integer j, every
v-chain in S
j+1 is O-dominated by xj,j+1.
Proof: First suppose that x dominates S. Let j be an odd integer and let A
a v-chain in S
j+1. We need to show that xj,j+1 dominates SA. For this,
we may assume that A is maximal (by Corollary 6.1.2). By Corollary 6.1.3 (3),
the length of A is at most 2. By Lemma 6.3.1 (5) (b), for every β in S
j+1 there
exists α in S
j with α > β. Thus we may assume that the head α of A belongs
It is enough to show (see [7, Lemma 4.5]) that for any v-chain E in SA
• the length of E is at most 2;
• there exists an x-dominated monomial in N containing E and the head
of E is an element of depth at least j in that monomial.
The first of these conditions holds by Proposition 7.5.4. We now show that the
second holds.
We may assume that E is maximal in SA. By Proposition 5.3.4 (1), the
head of E is qA,α. Let C a v-chain in S with tail α such that O-depthC(α) = j.
Let D be the concatenation of C with A. We claim that the monomial SD has
the desired properties. That SD is x-dominated is clear (since x O-dominates
S). By Corollary 6.3.5, it follows that qD,α = qA,α and SA ⊆ SD (in particular
that E ⊆ SD). By Proposition 6.1.1 (2), O-depthD(α) = O-depthC(α) = j,
that is, depth
qD,α = j. The proof of the necessity is thus complete.
To prove the sufficiency, proceed by induction on the largest odd integer J
such that S
J+1 is non-empty. When J = 1, there is nothing to prove,
for S
1 ∪ S
2 = S and x1,2 O-dominates S
1 ∪ S
2 . So suppose that J ≥
3. We implicitly use Corollary 6.3.7 in what follows. By induction, x3 O-
dominates S3,4.
Let D be a v-chain in S. Our goal is to show that x dominates SD. Let
α be the element of D with O-depthD(α) = 3—such an element exists, by
Lemma 6.3.1 (5) (if there exists in D an element of O-depth in D exceed-
ing 2); the following proof works also in the case when α does not exist. Let
A be the part α > . . . of D, and C′ the part up to but not including α. By
Proposition 6.1.1 (2), the O-depth (in C′) of elements of C′ is at most 2. By
Proposition 9.2.1, x1,2 dominates SC′ . By Corollary 6.3.5 (3), (SD)1,2 = SC′
and (SD)
3 = SA. Since A ⊆ S
3,4, it follows that x3 dominates SA (induction
hypothesis). Finally, by an application of Corollary 9.1.5, we conclude that x
dominates SD. �
Corollary 9.2.3 Let S be a a monomial in ON and x an element of I(d). For
x to O-dominate S it is necessary and sufficient that x1,2 O-dominate S
and x3 O-dominate S3,4.
Proof: It is easy to see that (x3)j,j+1 = xj+2,j+3; it follows from Proposi-
tion 6.3.6 that (S3,4)
j ∪ (S
j+1 = S
j+2 ∪S
j+3. The assertion follows from
the lemma. �
9.3 Orthogonal analogues of some lemmas in [7]
The proofs of Propositions 4.1 and 4.2 of [7] are based on assertion 4.9–4.16 (of
that paper). Assertion 4.9 being a statement about a single Sk, it is applicable
in the present situation. Since references to it are frequent, we recall it below
as Lemma 9.3.1. As to assertions 4.10–4.16 of [7], assertions 9.3.2, 9.3.4–9.3.9
below are their respective analogues.
A block of a monomial S in ON means a block of Sj,j+1 in the sense of [7]
for some odd integer j.
Caution: Considering S as a monomial in N, there is the notion of a “block”
of S as in [7], which has in fact been used in §9.1, and which is different from the
notion just defined. Both notions are used and it will be clear from the context
which is meant.
Throughout this section S denotes a monomial in ON and j an integer (not
necessarily odd).
Lemma 9.3.1 If B1, . . . ,Bl are the blocks in order from left to right of some
Sk, and w(B1) = (R1, C1), w(B2) = (R2, C2), . . ., w(Bl) = (Rl, Cl), then
C1 < R1 < C2 < R2 < . . . < Rl−1 < Cl < Rl
Proof: This is merely a recall Lemma 4.9 of [7]. In any case it follows easily
from the definitions. �
Lemma 9.3.2 No two elements of Sk(ext) ∪ S
k are comparable. More pre-
cisely, it is not possible to have elements α > β both belonging to Sk(ext)∪S
Proof: It follows from Lemma 9.3.1 that Sk ∪ S
k contains no comparable
elements. If k is even, then Sk(ext) = Sk (Corollary 7.3.4 (2)); if k is odd, we
may assumeSk(ext) = Sk (as sets) by increasing the multiplicity of σk in S
Lemma 9.3.3 For integers i ≤ k, there cannot exist γ ∈ S′i(up) and β ∈ S
such that β > γ. For integers i < k, there cannot exist γ ∈ S′i(up) and β ∈ S
such that β dominates γ.
Proof: Let γ ∈ S′i(up) and β ∈ S
. If i = k and β > γ, then we get a
contradiction immediately to Lemma 9.3.2. Now suppose that i < k and that
β dominates γ. Apply Corollary 6.3.4 (the notation of the corollary being sug-
gestive of how exactly to apply it). Let α be as in its conclusion. The chain
α > γ contradicts Lemma 9.3.2 in case i is odd and either Lemma 9.3.2 or
Proposition 7.5.4 in case i is even. �
Lemma 9.3.4 For (r, c) in S′, there exists a unique block B of S with (r, c)
in B′.
Proof: The existence is clear from the definition of S′. For the uniqueness,
suppose that B and C are two distinct blocks of S with (r, c) in both B′ and C′.
We will show that this leads to a contradiction.
Let i and k be such that B ⊆ Si and C ⊆ Sk. From Lemma 4.11 of [7] (of
which the present lemma is the orthogonal analogue) it follows that i 6= k, so
we can assume without loss of generality that i < k. By applying the involu-
tion # if necessary, we may assume that (r, c) ∈ S′i(up). Now there exists an
element (r, a) in C with a ≤ c (this follows from the definition of C′). Clearly
(r, a) ∈ S
. Taking β = (r, a) and γ = (r, c), we get a contradiction to
Lemma 9.3.3. �
Lemma 9.3.5 Let i < j be positive integers.
1. Given a block B of Sj, there exists a unique block C of Si such that
w(C) > w(B).
2. Given an element β in Sj(ext)∪S
j, there exists α in Si such that α > β.
Proof: (1): The assertion follows by applying Lemma 9.1.2 with B = B
and U = Si. We need to make sure however that the lemma can be applied.
More precisely, we need to check that for every β = (r, c) in B there exist
γ1(β) = (R1, C1) and γ2(β) = (R2, C2) in Si such that C
1 < c, C2 < R1, and
r < R2. We may assume β = β(up), for, if β = β(down), then β(up) also belongs
to Sj because Sj is symmetric, and we can set γ
1(β) = γ2(β(up))(down),
and γ2(β) = γ1(β(up))(down)—note that these two belong to Si since Si is
symmetric.
We consider three cases:
1. β belongs to S.
2. β = ph(σj−1) (in particular, j is even and S is truly orthogonal at j − 1).
3. β = pv(σj) (in particular, j is odd and S is truly orthogonal at j).
Define β′ to be β in case 1, σj−1 in case 2, and σj in case 3. Let C be a v-chain
in S with tail β′ and having the good property as in Proposition 6.3.3.
First suppose that there exists in C an element of O-depth i and denote it
by γ. If ph(γ) 6∈ N (this can happen only in case 1), then set γ
1(β) = γ2(β) = γ.
Now suppose ph(γ) ∈ N. Then γ ∈ Si except when γ = σi with i odd and σi
has multiplicity 1 in S. If γ ∈ Si, take γ
1(β) = γ and γ2(β) = γ# = γ(down);
if γ 6∈ Si, then take γ
1(β) = γ2(β) = pv(γ).
Now suppose that C has no element ofO-depth i. Then, by Lemma 6.3.1 (5),
i is even and there exists in C an element of O-depth i − 1. This element
of C is of type H by Lemma 6.3.1 (1), so S is truly orthogonal at i − 1. Set
γ1(β) = γ2(β) = ph(σi−1).
(2): This proof parallels the proof of (1) above. As in the above proof, we
may assume that β = β(up). Suppose β = (r, c) belongs to S′j. Then there
exists (r, a) ∈ Sj with a ≤ c. Since S
j does not meet the diagonal, it is clear
that (r, a) ∈ ON, and thus it is enough to prove the assertion for β ∈ Sj(ext).
So now take β ∈ Sj(ext). Let β
′ and C be in the proof of (1). First suppose
that there exists in C an element of O-depth i. Denote it by γ. If γ ∈ Si, then
take α = γ. If γ 6∈ Si, then pv(γ) ∈ Si, and we take α = pv(γ). In case there is
no element in C of O-depth i, we take α = ph(σi−1) (see the above proof). �
Corollary 9.3.6 If B and B1 are blocks of S with w(B) = (r, c) and w(B1) =
(r1, c1), then exactly one of the following holds:
c < r < c1 < r1, c1 < r1 < c < r,
c < c1 < r1 < r, or c1 < c < r < r1.
Proof: This is a formal consequence of Lemmas 9.3.1 and 9.3.5, just as Corol-
lary 4.13 of [7] is of Lemmas 4.9 and 4.12 of that paper. �
Corollary 9.3.7 If w(B) > w(C) for blocks B ⊆ Si and C ⊆ Sj of S, then
i < j.
Proof: This is a formal consequence of Lemmas 9.3.1 and 9.3.5. It follows
from the first lemma that i 6= j. Suppose i > j. Then there exists by the second
lemma a block C′ ⊆ Sj such that w(C
′) > w(B). But then w(C′) > w(C), a
contradiction of the first lemma. �
Corollary 9.3.8 Let (s, t) > (s1, t1) be elements of S
′, and B, B1 be blocks of
S such that (s, t) ∈ B′, and (s1, t1) ∈ B
1. Then w(B) > w(B1).
Proof: Let w(B) = (r, c) and w(B1) = (r1, c1). By Corollary 9.3.6, we have
four possibilities. Since (r, c) dominates (s, t) and (r1, c1) dominates (s1, t1),
the possibilities c < r < c1 < r1 and c1 < r1 < c < r are eliminated. It is
thus enough to eliminate the possibility c1 < c < r < r1. Suppose that this
is the case. Then, by Corollary 9.3.7, j1 < j, where j1 and j are such that
B ⊆ Sj and B1 ⊆ Sj1 . Now, by Lemma 9.3.5 (2), there exists α in Sj1 such
that α > (s, t) > (s1, t1). But then this contradicts Lemma 9.3.2. �
Corollary 9.3.9 For a B ⊆ Si of S, the depth of w(B) in Sw is exactly i.
Proof: That the depth is at least i follows from Lemma 9.3.5. That the depth
cannot exceed i follows from Corollary 9.3.7. �
Corollary 9.3.10 Let α ∈ S′k(up), β ∈ S
m(up), and α > β. Then k < m.
Proof: Corollary 9.3.8 and Corollary 9.3.9. �
9.4 More lemmas
This subsection is a collection of lemmas to be invoked in the later subsec-
tions. More specifically, Lemma 9.4.1 and Corollary 9.4.2 are invoked in the
proof of Proposition 4.1.1 in §10.1, Lemma 9.4.3 in the proof of the first half of
Proposition 4.1.2 in §10.2, and Lemma 9.4.4 in the proof of the second half of
Proposition 4.1.2 in §10.3. Throughout this subsection, S denotes a monomial
in ON.
Lemma 9.4.1 Let C be a v-chain in S′, α an element of C, and α′ ∈ SC,α.
Then depth
α′ ≤ k(even), for k the integer such that α ∈ S′k(up).
Proof: Proceed by induction on k. If k = 1, the assertion follows from Corol-
lary 9.3.10, so assume k > 1. Choose a v-chain C′ in SC with tail α
′ and
depthC′α
′ = depth
(α′). The length of a v-chain in SC,α is clearly at most 2.
So, if γ′ is the element two steps before α′ in C′ (if γ′ does not exist then there is
clearly nothing to prove), then γ′ ∈ SC,γ with γ > α (see Proposition 5.3.4 (2)).
We claim that depth
(γ′) ≤ k(odd) − 1. It is enough to prove the claim, for
then depth
(α′) = depthC′α
′ = depthC′γ
′ + 2 ≤ k(odd)− 1 + 2 = k(even).
The claim follows by induction from Corollary 9.3.10 if k is odd or more
generally if γ ∈ S′l(up) with l ≤ k(odd) − 1. So assume that k is even and
γ ∈ S′k−1(up). By 7.5.4, it is not possible that γ is of type H and ph(γ) > α.
So the only possibility is that α′ = ph(α) and γ > α is connected. In particular,
γ is of type V and α of type H in C and γ′ = pv(γ).
Now let µ be the first element in the connected component of α in C. The
cardinality of the part µ > . . . > γ of C is even (by Proposition 5.3.1 (1), it
follows that the cardinality of µ > . . . > α is odd), say e. Letting m be such
that µ ∈ S′m(up), we have, by Proposition 9.3.10, m ≤ k − 1 − (e − 1) =
k − e. If m(even) < k − e, then, since depthC′γ
′ = depthC′pv(µ) + e − 1 (by
Proposition 5.3.4 (1), since, by Proposition 5.3.1 (2), µ, . . . , γ all have type V
in C) and depthC′pv(µ) ≤ m(even) by induction, it follows that depthC′γ
k − e+ e− 1 = k − 1, and we are done.
So suppose that m(even) = k − e. Let ν be the element just before µ in C
(if such an element does not exist, then depthC′γ
′ = e ≤ k − 2—observe that
m(even) ≥ 2—and we are done). Then ν > µ is not connected (by choice of µ).
So ph(ν) > µ. By Proposition 7.5.4, this means that j ≤ m(even) − 2 where j
is the odd integer defined by ν ∈ S′j(up) ∪ S
j+1(up). So, again by induction,
depthC′γ
′ = depthC′ph(ν) + e ≤ m(even) − 2 + e = k − 2, and the claim is
proved. �
Corollary 9.4.2 The O-depth of an element α in S′ is at most k where k is
such that α ∈ S′k(up).
Proof: Let C′ be a v-chain in SC with tail qC,α. If k is even, then, by the
lemma, depthC′qC,α ≤ k. So suppose that k is odd. Let γ
′ be the immediate pre-
decessor of qC,α in C
′. By Proposition 5.3.4 (2), γ > α, and so γ ∈ S′l(up) with
l ≤ k− 1 (see the observation in the first paragraph of the proof of the lemma).
So depthC′γ
′ ≤ k − 1 (by the lemma) and depthC′α
′ = depthC′γ
′ + 1 ≤ k. �
Lemma 9.4.3 Let S be a monomial in ON and Oπ(S) = (w,S′). Let i <
k be integers, α an element of S′i(up), and δ an element of (Sw)k(up) that
dominates α.
1. If k is even, then there exists β ∈ S′k(up) with α > β.
2. If k is odd and wk,k+1 O-dominates the singleton v-chain α, then either
there exists β ∈ S′k(up) with α > β or there exists γ ∈ S
k+1(up) with
ph(α) > γ.
Proof: Write α = (r, c) and δ = (A,B). By Corollary 9.3.9, there exists
a block B of Sk such that δ = w(B). Let (D,B) be the first element of B
(arranged in increasing order of row and column indices). We have the following
possibilities:
(i) D ≤ A and (D,B) ∈ S
(ii) k is odd, S is truly orthogonal at k, (D,B) = (A,B) = pv(σk), and B
consists of the single diagonal element (D,B) = (B∗, B).
(iii) k is even, S is truly orthogonal at k−1, (D,B) = (A,B) = ph(σk−1), and
B consists of the single diagonal element (D,B) = (B∗, B).
We claim the following: in case (i), D < r (in particular, D < A); in case (ii),
the row index of σk is less than r; and case (iii) is not possible. The first
two assertions and also the third in the case i < k − 1 follow readily from
Lemma 9.3.3; in case (iii) holds and i = k − 1, then σk−1 > α, a contradiction
to Lemma 9.3.2.
First suppose that possibility (ii) holds. Write σk = (s,B). Since s < r
and ph(σk) ∈ N, it is clear that ph(α) = (r, r
∗) also belongs to N. From the
hypothesis that wk,k+1 O-dominates {α}, it follows that there is an element
of (Sw)k+1 that dominates ph(α) = (r, r
∗). Such an element must be diagonal
(because of the distinguishedness ofSw), and so must be the w(C) for the unique
diagonal block C ofSk+1. In particular, this means that there are elements other
than (s, s∗) in Sk+1, and so S
k+1 is non-empty. In the arrangement of elements
of S′k+1(up) in increasing order of row and column numbers, let γ = (e, s
∗) be
the last element. Then e < s < r and r∗ < s∗, so ph(α) > γ, and we are done.
Now suppose that possibility (i) holds. Let (p, q) be the element of Sk such
that p is the largest row index that is less than r, and, among those elements
with row index p, the maximum possible column index is q. The arrangement of
elements of Sk (in increasing order or row and column indices) looks like this:
. . . , (p, q), (s, t), . . .
Since p < r ≤ A and w(B) = (A,B), we can be sure that (p, q) is not the last
element of B.
We first consider the case c < t. Then α = (r, c) > β := (p, t) ∈ S′k.
If β ∈ S′k(up), then we are done. It is possible that (p, q) lies on or below
the diagonal so that β lies below the diagonal, in which case, α > β(up) and
β(up) ∈ S′k(up), and again we are done.
Now suppose that t ≤ c. We claim that:
• (s, t) belongs to the diagonal;
• k is odd and S is truly orthogonal at k; and
• σk = (u, t) with u < r.
Suppose that (s, t) does not belong to the diagonal. Since r ≤ s (by choice of
(p, q)), it follows that (s, t) dominates (r, c). This leads to a contradiction to
Lemma 9.3.3, for either (s, t) or its reflection (t∗, s∗) (whichever is above the
diagonal) belongs to S
and dominates α = (r, c) in S′i(up). This shows that
(s, t) belongs to the diagonal. If k is even, then (s, t) = ph(σk−1), which means
σk−1 > α, again contradicting Lemma 9.3.3, so k must be odd. It also follows
that S is truly orthogonal at k and that (s, t) = pv(σk). Writing σk = (u, t), if
r ≤ u, then σk would dominate α, again contradicting Lemma 9.3.3. So u < r,
and the claim is proved.
To finish the proof of the lemma, now proceed as in the proof when possi-
bility (ii) holds. �
Lemma 9.4.4 Let T be a monomial in ON and w an element of I(d) that O-
dominates T. Let β′ > β be elements Sw(up). Let d−1 and d be their respective
depths in Sw. Let α be an element of OP
β or more generally an element of
ON such that
(a) it is dominated by β,
(b) it is not comparable to any element of Pβ, and
(c) in case d is odd, then {α} ∪ Tw,d,d+1 has O-depth at most 2.
1. there exists α′ ∈ P∗β′(up) with α
′ > α;
2. for α′ as in (1), if α′ is diagonal, then ph(δd−2) > α if d is odd and
δd−1 > α if d is even.
Proof: Assertion (2) is rather easy to prove. If d is odd, then, in fact,
ph(δd−2) = α
′; if d is even, then δd−1 has the same column index as α
′ and, by
Proposition 8.2.2 (4), has row index more than that of α, so δd−1 > α.
Let us prove (1). Write α = (r, c), β = (R,C), and β′ = (R′, C′). There
exists, by the definition of P∗β′ , an element in P
β′ with column index C
′. We
have C′ < c (for C′ < C ≤ c). Let (r′, c′) be the element of P∗β′ such that c
is maximum possible subject to c′ < c and among those elements with column
index c′ the maximum possible row index is r′. If r < r′, then we are done (if
(r′, c′) is below the diagonal, its mirror image would have the desired properties).
It suffices therefore to suppose that r′ ≤ r and arrive at a contradiction.
In the arrangement of elements of P∗β′ in non-decreasing order of row and
column indices, there is a portion that looks like this:
. . . , (r′, c′), (a, b), . . .
Since there is in P∗β an element with row index R
′ (and clearly r′ ≤ r < R < R′),
it follows that (a, b) exists (that is, (r′, c′) is not the last element in the above
arrangement). It follows from the construction of P∗β′ from Pβ′ that (r
′, b) is an
element in Pβ′ . By the choice of (r
′, c′), we have c ≤ b. Thus (r, c) dominates
(r′, b).
The proof now splits into two cases accordingly as d is even or odd. First
suppose that d is even. Then, since β dominates (r′, b) and yet (r′, b) does not
belong to Pβ, there exists a v-chain in Tw,d−1,d of length 2 and head (r
′, b). The
tail of this v-chain then belongs toPβ and is dominated by (r, c), a contradiction
to our assumption that α is not comparable to any element of Pβ.
Now suppose that d is odd. Choose a v-chain C in T with head (r′, b) that
is not O-dominated by wd. Let D be the part of C consisting of elements of
O-depth (in C) at most 2. We claim that D is O-dominated by wd,d+1. In fact,
we claim the following: Any v-chain F with head (r′, b) and O-depth at most 2
is O-dominated by wd,d+1.
To prove the claim, we first prove the following subclaim:
(†) If the horizontal projection of (r′, b) belongs to N, then β is on
the diagonal and dominates the vertical projection of (r′, b), and the
diagonal element β1 of (Sw)d+1 dominates the horizontal projection
of (r′, b).
Let ph(r
′, b) ∈ N. Then β belongs to the diagonal because Sw is distinguished
and symmetric. Once β is on the diagonal, it is clear that it dominates pv(r
′, b)
(from our assumptions, β dominates (r, c) and (r, c) dominates (r′, b)). It follows
from Proposition 8.2.2 (3) that the row index of β1 exceeds the row index r of
(r, c), so β1 dominates ph(r
′, b). This finishes the proof of the subclaim (†).
To begin the proof of the claim, observe that F has length at most 2. Suppose
first that F consists only of the single element (r′, b). The type of (r′, b) in
F is either H or S. If it is S, then since β dominates (r′, b), the claim follows
immediately. If it is H, then the claim follows immediately from the subclaim (†).
Continuing with the proof of the claim, let now F consist of two elements:
(r′, b) > µ. Let γ be the element of Sw such that µ ∈ Pγ , and let e be the
depth of γ in Sw. From Lemma 8.2.1 (2b) it follows that e ≥ d. If e = d, then
γ = β (by the distinguishedness of Sw), and the comparability of (r, c) and µ
contradicts our hypothesis (b). So e ≥ d+1, and there exists δ of depth d+1 in
Sw that dominates µ. We have β > δ (again by the distinguishedness of Sw).
The possibilities for the types of (r′, b) and µ in F are: S and S, V and V, H
and S (in the last case ph(r
′, b) 6> µ by Lemma 6.3.1 (1)). Noting the existence
in (Sw)d,d+1 of the v-chain β > δ in the first case and also of β > β1 (where
β1 is as in the subclaim) in the last case, the proof of the claim in these cases
is over. So suppose that the second possibility holds. The distinguishedness of
Sw implies that δ = β1. Since δ is diagonal, it dominates the vertical projection
of µ. Noting the existence of the v-chain in β > δ in (Sw)d,d+1, the proof of
the claim in this case too is over.
We continue with the proof of the lemma. It follows from the claim that D is
O-dominated by wd,d+1. From Corollary 9.2.3 it follows that the complement E
of D in C is not O-dominated by wd+2,d+3 (in particular, that E is non-empty)
and that every v-chain in T with head ǫ (where ǫ denotes the head of E) is
O-dominated by wd (given such a v-chain, the concatenation of D with it is
O-dominated by wd−2, and ǫ continues to have O-depth 3 in the concatenated
v-chain). Thus ǫ belongs to Tw,d,d+1. From (1) and (2b) of Lemma 8.2.1 it
follows that the element µ of C in between (r′, b) and ǫ (if it exists at all) also
belongs to Tw,d,d+1. Now consider the v-chain obtained as follows: take the part
of C up to (and including) ǫ and replace its head (r′, b) by (r, c). This chain has
O-depth 3 and lives in {α} ∪ Tw,d,d+1, a contradiction to hypothesis (c). �
Corollary 9.4.5 Let T be a monomial in ON and w an element of I(d) that
O-dominates T. Let β′ > β be elements of Sw(up), α an element of OP
β, and
d′ := depth
1. If d′ is odd, there exists α′ ∈ OP∗β′ such that α
′ > α.
2. If there does not exist α′ ∈ OP∗β′ such that α
′ > α then (d′ is even by (1)
above and) there exists α′′ ∈ OP∗β′′ such that ph(α
′′) > α, where β′′ is the
unique element of (Sw)d′−1 such that β
′′ > β′.
Proof: Immediate from the lemma. �
Corollary 9.4.6 Let T be a monomial in ON and w an element of I(d) that
O-dominates T. Let β, β′ be elements of Sw(up), and α, α
′ elements of OP∗β
and OP∗β′ respectively.
1. If α′ > α then β′ > β (in particular, depth
β′ < depth
2. If ph(α
′) > α and depth
β is even, depth
β′ ≤ depth
β − 2.
Proof: (1) Writing β = (r, c) and β′ = (r′, c′), there are, since both β and β′
dominate α and Sw is distinguished, the following four possibilities:
c < r < c1 < r1, c1 < r1 < c < r, c < c1 < r1 < r, c1 < c < r < r1
Since α′ > α, and α, α′ are dominated respectively by β, β′ (this is because α,
α′ belong to OP∗β, OP
β′ respectively), the possibilities c < r < c1 < r1 and
c1 < r1 < c < r are eliminated (by the distinguishedness of Sw). It is thus
enough to eliminate the possibility β > β′. Suppose, by way of contradiction,
that β > β′. By Corollary 9.4.5, either there exists γ ∈ OP∗β such that γ > α
in which case the v-chain γ > α in OP∗β contradicts Proposition 8.2.2 (3) or (4),
or d := depth
β is even and there exists (with β′′ being the unique element
in Sw such that β
′′ > β and depth
β′′ = d− 1) an element α′′ ∈ OP∗β′′ with
′′) > α′, in which case the v-chain α′′ > α in T⋆w,d−1,d has O-depth 3 and
so contradicts Proposition 8.2.2 (2).
(2) Set d := depth
β. If depth
β′ were d− 1, then the v-chain α′ > α in
T⋆w,d−1,d would be ofO-depth 3 and so would contradict Proposition 8.2.2 (2). �
10 The Proof
The aim of this section is to prove Propositions 4.1.1 and 4.1.2. The proof of
first proposition appears in §10.1 and that of the second in §§10.2, 10.3. In §9.4
some lemmas are established that are used in the proofs. Needless to say that
the lemmas maybe unintelligible until one tries to read the proofs in the later
subsections.
10.1 Proof of Proposition 4.1.1
(1) By definition, w is the element of I(d) associated to the distinguished mono-
mial ∪kSw(k). By the very definition of this association, we have w ≥ v.
(2) This follows from the corresponding property of the map π of [7]. More
precisely, that property justifies the third equality below. The other equalities
are clear from the definitions.
v-degree(w) + degree(S′) =
degree(Sw) +
degree(S′k)
degree(Sw(k)) + degree(S
degree(Sk)
j odd
degree(Sj,j+1)
j odd
degree(S
j ) + degree(S
= degree(S)
(3) We have:
w O-dominates S′ ⇔ w ≥ wC ∀ v-chain C in S
⇔ w dominates SC ∀ v-chain C in S
⇔ ∀ v-chain C in S′, ∀ α′ = (r, c) ∈ SC ,
∃ β = (R,C) ∈ Sw with C ≤ c, r ≤ R,
and depth
β ≥ depth
The first equivalence above follows from the definition of O-domination, the
second from [7, Lemma 4.5], the third from Lemma 9.1.4.
Now let C be a v-chain in S′ and α′ = (r, c) in SC . We will show that
there exists β in Sw that dominates α and satisfies depthSwβ ≥ depthSCα
Let α be the element in C such that α′ ∈ SC,α, let k be such that α ∈ S
k(up),
and let B be the block of Sk such that α ∈ B
′. Writing α = (r1, c1) and
w(B) = (R1, C1), we have C1 ≤ c1 and r1 ≤ R1 straight from the definition of
w(B). By Corollary 9.3.9, depth
w(B) = k.
First suppose that w(B) dominates α′ (meaning C1 ≤ c and r ≤ R1).
If k ≥ depth
α′, we are clearly done; by Corollary 9.4.2, this is the case
when α′ = qC,α. So suppose that α is of type H, α
′ = ph(α), and that k <
depth
α′. By Lemma 9.4.1, depth
α′ ≤ k(even). It follows that k is odd
and depth
α′ = k + 1. By Corollary 7.5.3, S is truly orthogonal at k, which
means that Sk+1 has a diagonal block, say C. Note that w(C) dominates ph(σk)
which in turn dominates ph(α). Since depthSww(C) = k+1 by Corollary 9.3.9,
we are done.
Now suppose that w(B) does not dominate α′. Then B is non-diagonal and
α′ = pv(α). Since B is non-diagonal, ph(α) 6∈ N, and α cannot be of type H.
So α is of type V in C. It follows easily (see Proposition 5.3.1 (3)) that α is
the critical element in C, and and that last element in its connected component
in C; by Lemma 6.3.1 (4), O-depthC(α) = depthSC qC,α =: d is even. By
Proposition 5.3.1 (1), (2), the cardinality of the connected component of α in
C is even.
The immediate predecessor γ of α in C is connected to α (this follows from
what has been said above). It is of type V in C, ph(γ) belongs to N, and
depth
pv(γ) = d − 1 (see Lemma 6.3.1 (1)). Let ℓ be such that γ ∈ S
ℓ(up).
Let C be the block of Sℓ such that γ ∈ C
′. Since ph(γ) ∈ N, C is diagonal.
Note that w(C) dominates pv(γ) and that pv(γ) > pv(α). By Corollary 9.3.9,
depth
w(C) = ℓ. Thus if d ≤ ℓ we are done. On the other hand, d− 1 ≤ ℓ by
Corollary 9.4.2.
So we may assume that ℓ = d− 1. By Corollary 7.5.3, S is truly orthogonal
at d − 1. This implies that Sd has a diagonal block, say D. Note that w(D)
dominates ph(σd−1) which in turn dominates ph(γ). Writing γ = (r2, c2), since
γ > α is connected, it follows that (r1, r
2) belongs to ON. Now both w(B) and
w(D) dominate (r1, r
2). Since Sw is distinguished and symmetric and w(B) is
not on the diagonal, it follows that w(D) > w(B). This implies, since w(D) is
on the diagonal, w(D) > pv(α). Since depthSww(D) = d by Corollary 9.3.9, we
are done.
(4) Let x be an element of I(d) that O-dominates S. We will show that
x ≥ w. By [7, Lemma 5.5], it is enough to show that x dominates Sw. By
Lemma 9.1.4, it is enough to show the following: for every block B of S, there
exists β in Sx such that β dominates w(B) and depthSxβ ≥ depthSww(B).
Let B be a block of S. By Corollary 9.3.9, depth
w(B) = k where B ⊆
Sk. Let S
x denote the set of elements of Sx of depth at least k. Our goal is
to show that there exists β in Skx that dominates w(B). It follows easily from
the distinguishedness of Sx and the fact that B is a block, that it suffices to
show the following: given α ∈ B, there exists β in Skx (depending upon α) that
dominates α. Moreover, since B and Skx are symmetric, we may assume that
α = α(up).
So now let α = α(up) belong to B. Then either
1. α belongs to S
2. k is odd, S is truly orthogonal at k, and α = pv(σk), or
3. k is even, S is truly orthogonal at k − 1, and α = ph(σk−1).
The proofs in the three cases are similar. In the first case, choose a v-chain C
in S with tail α such that O-depthC(α) = k (see Corollary 6.1.3 (1)). Then
depth
qC,α = k and, clearly, qC,α dominates α. Since x dominates SC , there
exists, by Lemma 4.5 of [7], β in Skx that dominates qC,α (and so also α).
In the second case, choose a v-chain C in S with tail σk with the property
that O-depthC(σk) = k. Then depthSC qC,σk = k. Since ph(σk) belongs to N,
σk is of type V or H in C, so qC,σk = α. Since x dominates SC , there exists, by
[7, Lemma 4.5], β in Skx that dominates qC,σk = α.
In the third case, choose a v-chain C in S with tail σk−1 such that the
O-depth in C of σk−1 is k − 1. Then depthSCqC,σk−1 = k − 1. Since ph(σk−1)
belongs to N, σk−1 is of type V or H in C, so qC,σk−1 = pv(σk−1). From
Lemma 6.3.1 (4), it follows, since k − 1 is odd, that σk−1 is of type H. Since
pv(σk−1) > ph(σk−1) = α, it follows that depthSCph(σk−1) ≥ k (in fact equality
holds as is easily seen). Since x dominates SC , there exists, by [7, Lemma 4.5],
β in Skx that dominates ph(σk−1) = α. �
10.2 Proof that OφOπ = identity
LetS be a monomial inON and let Oπ = (w,S′). We need to show thatOφ ap-
plied to the pair (w,S′) gets us back toS. We know from (3) of Proposition 4.1.1
that w O-dominates S′, so Oφ can indeed be applied to the pair (w,S′).
The main ingredients of the proof are the corresponding assertion in the
case of Grassmannian [7, Proposition 4.2] and the following claim which we will
presently prove:
(S′)w,j,j+1 = S
j(up) ∪S
j+1(up) for every odd integer j
Let us first see how the assertion follows assuming the truth of the claim, by
tracing the steps involved in applying Oφ to (w,S′). From the claim it follows
that when we partition S′ into pieces (see §8), we get S′j(up) ∪S
j+1(up) (for
odd integers j). Adding the mirror images will get us to S′j ∪ S
j+1. From
Corollary 9.3.9 it follows that wj,j+1 is exactly the element of I(d, 2d) obtained
by acting π on Sj ∪ Sj+1. Now, since φ ◦ π = identity, it follows that on
application of φ to (wj,j+1,S
j+1) we obtain Sj ∪Sj+1. By twisting the
two diagonal elements in Sj ∪ Sj+1 (if they exist at all) and removing the
elements below the diagonal d, we get back S
j,j+1. Taking the union of S
j,j+1
(over odd integers j), we get back S.
Thus we need only prove the claim. Since S′ is the union over all odd
integers of the right hand sides (this follows from the definition of S′), and the
left hand sides as j varies are mutually disjoint, it is enough to show that the
right hand side is contained in the left hand side. Thus we need only prove: for
j an odd integer and α an element in S′j(up) ∪S
j+1(up),
• every v-chain in S′ with head α is O-dominated by wj .
• there exists a v-chain in S′ with head α that is not O-dominated by wj+2.
To prove the first item, write T = Sj,j+1 := {α ∈ S|O-depth
(α) ≥ j}
and set Oπ(T) = (x,T′). By Proposition 6.3.6, we have T
i ∪ T
i+1 = S
i+j−1 ∪
i+j for any odd integer i. Thus, by the description of Oπ, we have T
∪k≥jS
k(up). By Corollary 9.3.9 and the description of Oπ, we have x = w
By Corollary 9.3.10, any v-chain inS′ with head belonging toS′j(up)∪S
j+1(up)
is contained entirely in ∪k≥jS
k(up). Finally, by Proposition 4.1.1 (3) applied
to T, the desired conclusion follows.
To prove the second item we use Lemma 9.4.3. Proceed by decreasing induc-
tion on j. For j sufficiently large the assertion is vacuous, for S′j(up)∪S
j+1(up)
is empty. To prove the induction step, assume that the assertion holds for j+2.
If the v-chain consisting of the single element α is not O-dominated by wj+2,
then we are done. So let us assume the contrary. Since the O-depth of the
singleton v-chain α is at most 2, it follows from Lemma 9.2.2 that wj+2,j+3 O-
dominates the v-chain α. Apply Lemma 9.4.3 with k = j+2. By its conclusion,
either there exists β ∈ S′j+2(up) such that α > β or there exists γ ∈ S
j+3(up)
such that ph(α) > γ.
First suppose that a γ as above exists. By induction, there exists a v-chain
in S′—call it D—with head γ that is not O-dominated by wj+4. Let C be the
concatenation of α > γ and D. Since elements of D haveO-depth at least 3 in C
(Lemma 6.3.1 (1)), it follows from Corollary 9.2.3 that C is not O-dominated
by wj+2, and we are done.
Now suppose that such a γ does not exist. Then a β as above exists. If
α > β is not O-dominated by wj+2 we are again done. So assume the contrary.
Since the O-depth of β in α > β is at least 2, it follows that there exists an
element of (Sw)j+3 that dominates β. Applying Lemma 9.4.3 again, this time
with k = j + 3, we find γ′ ∈ S′j+3(up) such that β > γ
′. Arguing as in the
previous paragraph with γ′ in place of γ, we are done. �
10.3 Proof that OπOφ = identity
Let T be a monomial in ON and w an element of I(d) that O-dominates T. We
can apply Oφ to the pair (w,T) to obtain a monomial T⋆w in ON. We need to
show that Oπ applied to T⋆w results in (w,T). The main step of the proof is to
establish the following:
T⋆w,j,j+1 = (T
j,j+1 (10.3.1)
(for the meaning of the left and right sides of the above equation, see §8 and
§7 respectively). Assuming this for the moment let us show that Oπ ◦ Oφ =
identity.
We trace the steps involved in applying Oπ to T⋆w. From Eq. (10.3.1) it
follows that when we break up T⋆w according to the O-depths of its elements
as in §7, we get T⋆w,j,j+1 (as j varies over odd integers). The next step in the
application of Oπ is the passage from (T⋆w)
j,j+1 to (T
w)j,j+1. This involves
replacing σj by its projections and adding the mirror image of the remaining
elements of (T⋆w)
j,j+1. It follows from Proposition 8.2.2 (3) that σj = δj and so
(T⋆w)j,j+1 = (Tw,j,j+1 ∪ T
w,j,j+1)
⋆. The next step is to apply π to (Tw,j,j+1 ∪
w,j,j+1)
⋆. Since π is the inverse of φ (as proved in [7]), we have π((Tw,j,j+1 ∪
w,j,j+1)
⋆) = (wj,j+1,Tw,j,j+1). Since Sw and T are respectively the unions, as
j varies over odd integers, of (Sw)j,j+1 and Tw,j,j+1, we see that Oφ applied to
T⋆w results in (w,T).
Thus it remains only to establish Eq. (10.3.1). It is enough to show that the
left hand side is contained in the right hand side, for the union over all odd j of
either side is T⋆w and the right hand side is moreover a disjoint union. In other
words, we need only show that the O-depth in T⋆w of an element of T
w,j,j+1
is either j or j + 1. We will show, more precisely, that, for any element β of
Sw, the O-depth in T
w of any element of OP
β equals the depth in Sw of β.
Lemma 9.4.4 will be used for this purpose.
Let α be an element of OP∗β and set e := O-depthT⋆w (α). We first show,
by induction on d := depth
β, that e ≥ d. There is nothing to prove in case
d = 1, so we proceed to the induction step. Let β′ be the element of Sw of depth
d − 1 such that β′ > β. If there exists α′ in OP⋆β′ with α
′ > α, the desired
conclusion follows from Corollary 6.1.3 (3) and induction. Lemma 9.4.4 says
that such an α′ exists in case d is even. So suppose that d is odd and such an
α′ does not exist. The same lemma now says that ph(δd−2) > α, so the desired
conclusion follows from Lemma 6.3.1 (1).
We now show, by induction on e, that d ≥ e. There is nothing to prove
in case e = 1, so we proceed to the induction step. Let C be a v-chain in T⋆w
with tail α and having the good property of Proposition 6.3.3. Let α′ be the
immediate predecessor in C of α. Let β′ in Sw be such that α
′ ∈ OP∗β′ (we are
not claiming at the moment that β′ is unique although that is true and follows
from the assertion that we are proving, the distinguishedness of Sw, and the
fact that β′ dominates α′). It follows from Corollary 9.4.6 that β′ > β.
Let d′ := depth
β′. It follows from Corollary 6.1.3 (3) that e′ < e where
e′ := O-depth
(α′). We have, d ≥ d′+1 ≥ e′+1 ≥ (e−2)+1 = e−1, the first
equality being justified because β′ > β, the second by the induction hypothesis,
and the last by Lemma 6.3.1 (1). It suffices to rule out the possibility that
d = e − 1. So assume d = e − 1. Then d = d′ + 1 and d′ = e′ = e − 2. It
follows from (1) of Lemma 6.3.1 that the v-chain α′ > α has O-depth 3 and
from (3) of the same lemma that e′ is odd. But then we get a contradiction to
Proposition 8.2.2 (2) (α′ and α belong to T⋆w,d′,d′+1). The proof of Eq. (10.3.1)
is thus over. �
10.4 Proof of Proposition 4.1.3
Observe that the condition (‡) makes sense also for a monomial of N. By virtue
of belonging to I(d), v has f∗ as an entry. It follows from the description of
the bijection w ↔ Sw of §5.1.2 that for an element w of I(d) to satisfy (‡) it is
necessary and sufficient that Sw (equivalently all its parts Sw,j,j+1) satisfy (‡).
(1) Since T satisfies (‡), so do its parts Tw,j,j+1 and Tw,j,j+1 ∪ T
w,j,j+1
(adding the mirror image preserves (‡)). Since Sw,j,j+1 also satisfies (‡), it
follows from the description of the map φ of [7] (observe the passage from a
piece P to its “star” P∗) that the (Tw,j,j+1 ∪ T
w,j,j+1)
⋆ satisfy (‡). Since
the “twisting” involved in the passage from (Tw,j,j+1 ∪ T
w,j,j+1)
⋆ to T⋆w,j,j+1
involves only a rearrangement of row and column indices, it follows that the
T⋆w,j,j+1 satisfy (‡). Finally so also does their union T
(2) The parts S
j,j+1 of S clearly satisfy (‡). Therefore so do the Sj,j+1, for,
first of all, adding the mirror image preserves (‡), and then the removal of σj
and addition of its projections involves only a rearrangement of row and column
indices. It follows from description of the map π of [7] (observe the passage from
a block B to the pair (w(B),B′)) that both Sw,j,j+1 and S
j,j+1 satisfy (‡).
Finally, Sw and S
′ being the union (respectively) of Sw,j,j+1 and S
j,j+1(up),
they satisfy (‡). �
Part IV
An Application
As an application of the main theorem (Theorem 2.3.1), an interpretation of
the multiplicity is presented.
11 Multiplicity counts certain paths
Fix elements v, w in I(d) with v ≤ w. It follows from Corollary 2.3.2 that the
multiplicity of the Schubert variety X(w) in Md(V ) at the point e
v can be in-
terpreted as the cardinality of a certain set of non-intersecting lattice paths. We
first illustrate this by means of two examples and then justify the interpreta-
tion.
11.1 Description and illustration
The points of N can be represented, in a natural way, as the lattice points of
a grid. The column indices of the points of the grid are the entries of v and
the row indices are the entries of {1, . . . , 2d} \ v. In Figure 11.1.1 the points
of ON and those of the diagonal in N are shown (for the specific choice of v in
Example 11.1.1). The open circles represent the points of Sw(up), where Sw
is the distinguished monomial in N that is associated to w as in §5.1.2. From
each point β of Sw(up) we draw a vertical line upwards from β and let β(start)
denote the top most point of ON on this line. In case β is not on the diagonal,
draw also a horizontal line rightwards from β and let β(finish) denote the right
most point of ON on this line. In case β is on the diagonal, then β(finish) is
not a fixed point but varies subject to the following constraints:
• β(finish) is one step away from the diagonal (that is, it is of the form (r, c),
for some entry c of v, where r is the largest integer less than c∗ that is not
an entry of v);
• the column index of β(finish) is not less than that of β;
• if depth
β is odd, then the horizontal projection of β(finish) is the same
as the vertical projection of γ(finish) where γ is the diagonal element of
Sw of depth 1 more than that of β.
With v and w as in Example 11.1.1, we have β(start) = (6, 3) and β(finish) =
(9, 5) for β = (9, 3); β(start) = β(finish) = (21, 20) for β = (21, 20); β(start) =
(15, 11) for the diagonal element β = (36, 11); β(start) = (6, 1) for the diagonal
element β = (46, 1). In the particular case (of non-intersecting lattice paths)
drawn in Figure 11.1.1, β(finish) = (27, 19) for β = (36, 11) and β(finish) =
(28, 14) for β = (46, 1).
A lattice path between a pair of such points β(start) and β(finish) is a se-
quence α1, . . . , αq of elements of ON with α1 = β(start) and αq = β(finish)
such that, for 1 ≤ j ≤ q − 1, if we let αj = (r, c), then αj+1 is either (R, c) or
(r, C) where R is the least element of {1, . . . , 2d} \ v that is bigger than r and
C the least element of v that is bigger than c. Note that if β(start) = (r, c) and
β(finish) = (R,C), then q equals
|({1, . . . , 2d} \ v) ∩ {r, r + 1, . . . , R}|+ |v ∩ {c, c+ 1, . . . , C}| − 1,
where | · | is used to denote cardinality.
Consider the set Pathsw of all tuples (Λβ)β∈Sw(up) of paths where
• Λβ is a lattice path between β(start) and β(finish) (if β is on the diagonal,
then β(finish) is allowed to vary in the manner described above);
• Λβ and Λγ do not intersect for β 6= γ.
The number of such p-tuples, where p := |Sw(up)|, is the multiplicity of X(w)
at the point ev.
Example 11.1.1 Let d = 23,
v = (1, 2, 3, 4, 5, 11, 12, 13, 14, 19, 20, 22, 23, 26, 29, 30, 31, 32, 37, 38, 39, 40, 41),
w = (4, 5, 9, 10, 14, 17, 18, 21, 23, 25, 27, 28, 31, 32, 34, 35, 36, 39, 40, 41, 44, 45, 46),
so that
Sw ={(9, 3), (10, 2), (17, 13), (18, 12), (21, 20), (25, 22), (27, 26),
(28, 19), (34, 30), (35, 29), (36, 11), (44, 38), (45, 37), (46, 1)}
and Sw(up) =
{(9, 3), (10, 2), (17, 13), (18, 12), (21, 20), (25, 22), (28, 19), (36, 11), (46, 1)}.
A particular element of Pathsw is depicted in Figure 11.1.1. �
Example 11.1.2 Figure 11.1.2 shows all the elements of Pathsw in the follow-
ing simple case:
d = 7, v = (1, 2, 3, 4, 7, 9, 10), and w = (4, 6, 7, 10, 12, 13, 14).
We have Sw = {(6, 3), (12, 9), (13, 2), (14, 1)}, Sw(up) = {(6, 3), (13, 2)(14, 1)}.
There are 15 elements in Pathsw and thus the multiplicity in this case is 15. �
Example 11.1.3 Let d = 10,
v = (1, 2, 3, 4, 6, 8, 11, 12, 14, 16), and w = (8, 9, 11, 14, 15, 16, 17, 18, 19, 20).
so that Sw = {(20, 1)(19, 2)(18, 3), (17, 4), (9, 6)(15, 12)}. Figure 11.1.3 shows
a tuple of paths that is disallowed (meaning one that is not in Pathsw). The
elements of ON are represented as usual by a grid. The slanted line represents
1 2 3 4 5
11 12 13 14
19 20
22 23
Figure 11.1.1: An element of Pathsw with v and w as in Example 11.1.1
the diagonal d. The solid dot represents the point of Sw(up) that is not on d,
and the crosses on d represent the points of Sw(up) that lie on d. The tuple is
disallowed because the horizontal projection of the last point of the path Λβ1
is not the vertical projection of the last point of the path Λβ2 , where β1 =
(20, 1) and β2 = (19, 2) are the diagonal elements of Sw of depths 1 and 2
respectively. �
11.2 Justification for the interpretation
We now justify the interpretation in the previous subsection of the multiplicity.
Corollary 2.3.2 says that the multiplicity is the number of monomials in OR
of maximal cardinality that are square-free and O-dominated by w. Any such
monomial contains OR \ON, for, by the definition of O-domination, adding or
removing elements of OR\ON to or from a monomial does not alter the status
of its O-domination. One could therefore equally well consider the number of
monomials in ON of maximal cardinality that are square-free and O-dominated
✲ ✲ ✲ ✲ ✲
❄ ❄ ❄ ❄ ❄
❄ ❄ ❄ ❄ ❄
❄ ❄ ❄
Figure 11.1.2: All the 15 non-intersecting lattice paths of Example 11.1.2
diagonal
1        2        3        4        6        8        11      12      14      16     
The horizontal projection of the last point of the path associated to (20,1) is
not the same as the vertical projection of the last point of the path associated
to (19,2).    This is the reason that this tuple is disallowed.
Figure 11.1.3: A disallowed tuple of lattice paths (see Example 11.1.3)
by w. We now establish a bijection between the set Monw of such monomials
and the set Pathsw of non-intersecting lattice paths as in §11.1.
Each element Λ of Pathsw can be thought of, in the obvious way, as a
monomial in ON. We will continue to denote the corresponding monomial by Λ.
It is clear that the monomial Λ is square-free and that all such monomials Λ
have the same cardinality (in particular, that if Λ1 ⊆ Λ2 for two such monomials
then Λ1 = Λ2). In order to establish the bijection it therefore suffices to prove
the following proposition.
Proposition 11.2.1 1. w is the element of I(d) obtained on application of
Oπ to the monomial Λ (in particular (see Proposition 4.1.1), the monomial
Λ is O-dominated by w).
2. Given a monomial T of ON that is square-free and O-dominated by w,
there exists Λ such that T ⊆ Λ.
Proof: (1) Write Λ = (Λβ)β∈Sw(up). From the description of the map Oπ
in §7, it follows that it suffices to show that Λ
(in the notation of §7) is the
union ∪Λβ where β runs over all elements of depth k in Sw(up). In other words,
it suffices to show that the O-depth in Λ of any element of Λβ equals the depth
in Sw of β. To prove this, we observe the following (these assertions are easily
seen to be true thinking in terms of pictures): for fixed β ∈ Sw(up) and α ∈ Λβ ,
(A) For β′ in Sw(up) such that β
′ > β, there exists α′ ∈ Λβ′ such that α
′ > α.
(B) If α′ > α for some α′ in Λβ′ for some β
′ in Sw(up), then β
′ > β. If,
furthermore, β and β′ are diagonal, their depths in Sw are 1 apart, and
the depth in Sw of β is even, then the following is not possible: ph(α
belongs to N and ph(α
′) > α.
From (A) it is immediate that the O-depth e in Λ of an element α of Λβ is not
less than the depth d in Sw of β. We now show, by induction on e, that e ≤ d.
For e = 1 there is nothing to show. Suppose that e ≥ 2. Let C be a v-chain
in Λ having tail α and the good property of Proposition 6.3.3, α′ the immediate
predecessor in C of α, e′ the O-depth of α′ in Λ, β′ the element of Sw(up)
such that α′ ∈ Λβ′ , and d
′ the depth in Sw of β
′. From Corollary 6.1.3 (3) it
follows that e′ ≤ e − 1, so we may apply induction. From (B) it follows that
d′ ≤ d − 1, so that, by induction, e′ ≤ d − 1. If e′ ≤ d − 2, then we are done
by Lemma 6.3.1 (1). So suppose that e′ = d′ = d − 1. If d is odd, then the
conclusion e ≤ d follows from (1) and (3) of the same lemma. In case d is even,
then it follows from condition (B) and (1) of the same lemma.
(2) Let T be a square-free monomial in ON that is O-dominated by w. To
construct Λ such that T ⊆ Λ, we construct the “components” Λβ . As in §8, let
Pβ denote the piece of T corresponding to β ∈ Sw. From every point belonging
to Pβ(up) and also from β(start) carve out the South-West quadrant; if β is
not diagonal, then do this also from β(finish). The boundary of the carved out
portion (intersected with ON) gives a lattice path starting from β(start). In
case β is not diagonal, the path ends in β(finish). In this case as well as in
the case when β is diagonal and of even depth in Sw, we take Λβ to be this
lattice path. In case β is diagonal and of odd depth in Sw we do the carving
out from one more point before taking Λβ to be the boundary of the carved
out region, namely from the point that is one step away from the diagonal and
whose horizontal projection is the vertical projection of the end point of Λγ
where γ is the diagonal element of Sw of depth 1 more than β. We need to
justify why carving out from the extra point is still valid, and we do this now
by applying Lemma 8.1.4.
Let us first choose notation that is consistent with that of that lemma. Let
β and γ be diagonal elements in Sw of depths d and d + 1. Assume that d is
odd. Let the pieces of T corresponding to β and γ, when their elements are
arranged in increasing order of row and column indices, look like this:
. . . , (r1, a
∗), (a, r∗1), . . . ; . . . , (r2, b
∗), (b, r∗2), . . .
It is easy to see that the conditions on the numbers in the above display that pro-
vide the requisite justification are: r1 ≤ b and a
∗ < b∗ (if Pβ is empty then the
justification is easy). To prove that a∗ < b∗, observe that the diagonal elements
in P∗β and P
γ are respectively (a, a
∗) and (b, b∗), and apply Lemma 8.1.3 (2).
That r1 ≤ b now follows from Lemma 8.1.4 (1). This finishes the justification.
It suffices to prove the following claim: the lattice paths Λβ as β varies are
non-intersecting. Suppose that Λβ and Λβ′ intersect for β 6= β
′. Let α be a point
of intersection. Clearly β dominates all elements of Λβ and in particular α; for
the same reason β′ also dominates α. By the distinguishedness of Sw, we may
assume without loss of generality that β′ > β. It is easy to see graphically that
if γ in Sw is such that β
′ > γ > β then Λγ intersects either Λβ′ or Λβ : consider
the open portion of ON “caught between” the segment of Λβ′ from β
′(start) to
α and the segment of Λβ from β(start) to α; the starting point γ(start) of Λγ
lives in this region but its ending point does not (points strictly to the Northwest
of α can neither be of the form γ(finish) for γ not on the diagonal nor can they
be one step away from the diagonal); so Λγ must intersect one of the two lattice
path segments. We may therefore assume that the depths of β′ and β differ
by 1.
We now apply Lemma 9.4.4. From the construction of Λβ it readily fol-
lows that α satisfies the hypotheses (a), (b), and (c) of that lemma. By the
conclusion of Lemma 9.4.4, there exists α′ ∈ P∗β′(up) such that α
′ > α. On
the other hand, it follows from the construction of P∗β′ from Pβ′ , and from the
construction of Λβ′ that two elements one from P
β′ and another from Λβ′ are
not comparable. This is a contradiction to the comparability of α′ and α. �
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Index
>, relation on ON . . . . . . . . . . . . . . . . . 11
≤, partial order on I(d, 2d) . . . . . . . . 10
A, affine patch qv 6= 0 of Md(V ) . . . 14
α(down), for α ∈ N . . . . . . . . . . . . . . . . 25
α# for α in N . . . . . . . . . . . . . . . . . . . . . 22
α(up), for α ∈ N . . . . . . . . . . . . . . . . . . .25
anti-domination . . . . . . . . . . . . . . . . . . . 18
B, a specific Borel subgroup . . . . . . . . 8
β(finish), for β ∈ Sw(up) . . . . . . . . . . 65
β(start), for β ∈ Sw(up) . . . . . . . . . . . 65
block
in the sense of [7] . . . . . . . . . . . . .47
of a monomial S in ON . . . . . . . 51
comparability, of elements of R . . . . 29
connected components of a v-chain.23
connectedness of two succcessive ele-
ments in a v-chain . . . . . . . . 22
critical element (of a v-chain) . . . . . . 24
d, integral part of n/2 (unfortunately
also used otherwise) . . . . . 7, 9
degree, of a monomial . . . . . . . . . . . . . 10
degree, of a standard monomial . . . . 16
δj , for j odd . . . . . . . . . . . . . . . . . . . . . . . 44
depth (of an element α in a monomial
S in N) = depth
α . . . . . . 46
depth (of a monomial S in N) . . . . . 46
diagonal, d . . . . . . . . . . . . . . . . . . . . . 10, 11
distinguished (a subset of N) . . . . . . 21
domination (among elements of R) 29
domination map . . . . . . . . . . . . . . . . . . . 18
e1, . . . , en, a specific basis of V . . . . . . 8
ev, T -fixed point . . . . . . . . . . . . . . . . . . . 10
〈 , 〉, bilinear form on V . . . . . . . . . . . . 7
fθ := qθ/qv . . . . . . . . . . . . . . . . . . . . . . . . 14
head, of a v-chain. . . . . . . . . . . . . . . . . .11
horizontal projection ph(α) . . . . . . . . 22
I(d) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
I(d, 2d) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
i(even), for an integer i . . . . . . . . . . . . 29
i(odd), for an integer i . . . . . . . . . . . . . 29
intersection (of a monomial in a set
with a subset) . . . . . . . . . . . . 11
isotropic subspace . . . . . . . . . . . . . . . . . . 7
In . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
I ′n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
k, base field, (characteristic 6= 2) .7, 15
k∗(:= n+ 1− k) . . . . . . . . . . . . . . . . 7, 10
L, line bundle . . . . . . . . . . . . . . . . . . . . . 13
Λβ , for β ∈ Sw(up) . . . . . . . . . . . . . . . . 66
lattice path, from β(start) to β(finish),
denoted Λβ . . . . . . . . . . . . . . . 65
legs of α, for α ∈ ON . . . . . . . . . . . . . . 22
legs, intertwining of . . . . . . . . . . . . . . . .22
length, of a v-chain . . . . . . . . . . . . . . . . 11
Md(V ), orthogonal Grassmannian 7, 8
Md(V )
′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
monomial. . . . . . . . . . . . . . . . . . . . . . . . . .10
w . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
multiplicity, of X(w) at ev . . . . . . . . . 65
multiset := monomial . . . . . . . . . . . . . . 10
N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
n := dimV , (even from §2.1 on) . . 7, 9
O(V ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
O-depth . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
ON . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10
OP∗β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Oφ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17, 42
Oπ . . . . . . . . . . . . . . . . . . . . . . . . . 17, 34, 35
OR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10
O-domination . . . . . . . . . . . . . . . . . . . . . 11
orthogonal Grassmannian (Md(V )) . 7
Paths
w . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
Pβ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
P∗β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Pfaffian qθ . . . . . . . . . . . . . . . . . . . . . . . . . 14
ph(α), horizontal projection. . . . . . . .22
piece of T (see also caution) . . . . . . . 43
pθ, Plücker coordinate . . . . . . . . . . . . . 13
pv(α), vertical projection . . . . . . . . . . 22
qC,α, for α in a v-chain C . . . . . . . . . . 24
qθ, Pfaffian . . . . . . . . . . . . . . . . . . . . . . . . 14
R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
S, fixed monomial in ON in §7, §9.3 .
S, set of monomials in OR . . . . . . . . 18
modifications . . see Notation 4.2.1
SC , where C is a v-chain . . . . . . . . . . 23
SC,α, for α in a v-chain C . . . . . . . . . 24
Schubert varieties . . . . . . . . . . . . . . . . 7, 8
S(down), for a monomial S . . . . . . . 25
S#, for monomial S in N or R . . . . 22
σk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Sj,j+1, for S in ON, j odd . . . . . . . . 34
Sj,j+1(ext), for S in ON, j odd . . . 38
j,j+1, for S in ON, j odd . . . . . . . . 34
Sk, for monomial S in N . . . . . . . . . . 46
Sk, for monomial S in ON . . . . . . . . 34
Sk(ext), for monomial S in ON . . . 39
Sk,k+1, for monomial S in N . . . . . . 49
, for monomial S in ON . . . . . . . 34
SM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
modifications . . see Notation 4.2.1
SMv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
SMw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
SO(V ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
S′, for monomial S in ON . . . . . . . . 35
standard monomial . . . . . . . . . . . . . . . . 14
v-compatible . . . . . . . . . . . . . . . . . . 15
w-dominated . . . . . . . . . . . . . . . . . . 14
S(up), for a monomial S . . . . . . . . . . 25
Sj,j+1, for monomial S in ON . . . . 32
Sk, for monomial S in N . . . . . . . . . . 46
Sw(v)(m) . . . . . . . . . . . . . . . . . . . . . . . . . .11
Sw, w in I(d, 2d) or I(d, n) . . . . 21, 48
Sw,j,j+1, w in I(d), j odd . . . . . . . . . 42
Sjw, w in I(d), j odd . . . . . . . . . . . . . . 42
symmetric (monomial of N). . . . . . . .22
T , a specific maximal torus . . . . . . . . . 8
T , set of monomials in ON . . . . . . . . 18
modifications . . see Notation 4.2.1
tail, of a v-chain . . . . . . . . . . . . . . . . . . . 11
T, fixed monomial in ON in §8 . . . . . . .
truly orthogonal at j (j odd) . . . . . . 34
Tw,j,j+1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
(Tw,j,j+1 ∪ T
w,j,j+1)
⋆ . . . . . . . . . . . . . . 43
T⋆w,j,j+1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
T⋆w(:= Oφ(w,T)) . . . . . . . . . . . . . . . . . . 44
type (V, H, S), of an element in a v-
chain. . . . . . . . . . . . . . . . . .23–24
U , set of monomials in OR \ON . . 18
modifications . . see Notation 4.2.1
u∗, for u ∈ I(d) . . . . . . . . . . . . . . . . . . . . 19
V , vector space of dimension n . . . . . .7
v, fixed element of I(d) . . . . . . . . . . . . 10
v-chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
v-degree . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
vertical projection pv(α) . . . . . . . . . . . 22
w(C) (or wC), where C is v-chain. .11
w#, for w an element of I(d, 2d) . . . 21
w(k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34
(w,S′)(:= Oπ(S)), for S in ON . . 34,
w∗, for w an element of I(d, 2d) . . . .21
wj,j+1, j odd . . . . . . . . . . . . . . . . . . . . . . 42
wj , j odd . . . . . . . . . . . . . . . . . . . . . . . . . . 42
X(w), Schubert variety . . . . . . . . . . . . 10
xk, x
k, xk,k+1, for x ∈ I(d, n). . .48–49
Xr,c, variable . . . . . . . . . . . . . . . . . . . . . . 16
Y (w)(:= X(w) ∩ A). . . . . . . . . . . . . . . .15
