# Initial Ideals of Tangent Cones to the Richardson Varieties in the Orthogonal Grassmannian

Authors: Shyamashree Upadhyay
Journal: International Journal of Combinatorics
 (2013)

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## Abstract

A Richardson varietyXαγ in the Orthogonal Grassmannian is defined to be the intersection of a Schubert variety Xγ in the Orthogonal Grassmannian and an opposite Schubert variety Xα therein. We give an explicit description of the initial ideal (with respect to certain conveniently chosen term order) for the ideal of the tangent cone at any T-fixed point of Xαγ, thus generalizing a result of Raghavan and Upadhyay (2009). Our proof is based on a generalization of the Robinson-Schensted-Knuth (RSK) correspondence, which we call the Orthogonal-bounded-RSK (OBRSK).


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## Body

## 1. Introduction

The Orthogonal Grassmannian is defined in Section2. A Richardson variety Xαγ in the Orthogonal Grassmannian (The Richardson varieties in the ordinary Grassmannian are also studied by Stanley in [1], where these varieties are calledskew Schubert varieties. Discussion of these varieties in the ordinary Grassmannian also appears in [2].) is defined to be the intersection of a Schubert variety Xγ in the Orthogonal Grassmannian with an opposite Schubert variety Xα therein. In particular, the Schubert and opposite Schubert varieties are special cases of the Richardson varieties. In this paper, we provide an explicit description of the initial ideal (with respect to certain conveniently chosen term order) for the ideal of the tangent cone at any T-fixed point eβ of Xαγ. It should be noted that the local properties of the Schubert varieties at T-fixed points determine their local properties at all other points, because of the B action; but this does not extend to the Richardson varieties, since Richardson varieties only have a T-action.In Raghavan and Upadhyay [3], an explicit description of the initial ideal (with respect to certain conveniently chosen term orders) for the ideal of the tangent cone at any T-fixed point of a Schubert variety in the Orthogonal Grassmannian has been obtained. In this paper, we generalize the result of [3] to the case of the Richardson varieties in the Orthogonal Grassmannian.Sturmfels [4] and Herzog and Trung [5] proved results on a class of determinantal varieties which are equivalent to the results of [6–8] for the case of the Schubert varieties (in the ordinary Grassmannian) at the T-fixed point eid. The key to their proofs was to use a version of the Robinson-Schensted-Knuth correspondence (which we will call the “ordinary” RSK) in order to establish a degree-preserving bijection between a set of monomials defined by an initial ideal and a “standard monomial basis.” The difficulty in extending this method of proof to the case of the Schubert varieties (in the ordinary Grassmannian) at an arbitrary T-fixed point eβ lies in generalizing this bijection, which is done in the three papers [6–8]; the work done in [8] is slightly more general, since it applies to the Richardson varieties, and not just to the Schubert varieties. These three bijections, when restricted to the Schubert varieties in the ordinary Grassmannian, are in fact the same bijection (This supports the conviction of the authors in [6] that this bijection is natural and that it is in some sense the only natural bijection satisfying the required geometric conditions.), although this is not immediately apparent. In the work of Kreiman in [8], this “generalized bijection” has been viewed for the first time as an extended version of the “ordinary” RSK correspondence, which he calls the Bounded-RSK correspondence; this viewpoint was not present in [6, 7]. Although the formulations of the bijections in [6, 7] are similar to each other, the formulation of the bijection in [8] is in terms of different combinatorial indexing sets.The work done in [3] by Raghavan and Upadhyay does not involve any version of the RSK correspondence, unlike the work done by Herzog and Trung in [5]. The work done in [3] relies on a degree-preserving bijection between a set of monomials defined by an initial ideal and a “standard monomial basis,” and this bijection is proved by Raghavan and Upadhyay in [9]. It is mentioned in [9] that it will be nice if the bijection proved therein can be viewed as a kind of  “Bounded-RSK” correspondence, as done by Kreiman in [8] for the case of the Richardson varieties in the ordinary Grassmannian. This paper fulfills the expectation of [9] that one should be able to view the bijection there as a generalized-bounded-RSK correspondence, which we call here the Orthogonal-bounded-RSK correspondence (OBRSK). Kreiman has mentioned in his paper [8] that he believes in the possibility of adapting the methods of his paper ([8]) to the Richardson varieties in the Symplectic and the Orthogonal Grassmannian as well; this paper supports Kreiman’s belief for the Orthogonal Grassmannian case.


## 2. The Orthogonal Grassmannian and Richardson Varieties in It

Fix an algebraically closed field𝔨 of characteristic not equal to 2. Fix a natural number d, a vector space V of dimension 2d over 𝔨, and a nondegenerate symmetric bilinear form 〈,〉 on V. For k an integer such that 1≤k≤2d, set k*:=2d+1-k. Fix a basis e1,…,e2d of V such that 〈ei,ek〉 equals 1 if i=k* and is 0 otherwise. Denote by SO(V) the group of linear automorphisms of V that preserve the bilinear form 〈,〉 and also the volume form. A linear subspace of V is said to be isotropic, if the bilinear form 〈,〉 vanishes identically on it. Denote by 𝔐d(V)′ the closed subvariety of the Grassmannian of d-dimensional subspaces consisting of the points corresponding to maximal isotropic subspaces. The action of SO(V) on V induces an action on 𝔐d(V)′. There are two orbits for this action. These orbits are isomorphic: acting by a linear automorphism that preserves the form but not the volume form gives an isomorphism. We denote by 𝔐d(V) the orbit of the span of e1,…,ed and call it the (even) Orthogonal Grassmannian. One can define the Orthogonal Grassmannian in the case when the dimension of V is not necessarily even. But it is enough to consider the case when the dimension of V is even, the reason being the following (see also Section 2.1 of [9] for the same): suppose that the dimension of V is odd; say dimension of V=2d+1. Let n~:=2d+2 and V~ be a vector space of dimension n~ with a nondegenerate symmetric form. Let e~1,…,e~n~ be a basis of V~ such that 〈e~i,e~k〉 equals 1 if i=k* and is 0 otherwise. Put e:=e~d+1 and f:=e~d+2. Take λ to be an element of the field such that λ2=1/2. We can take V to be the subspace of V~ spanned by the vectors e~1,…,e~d,λe+λf,e~d+3,…, ande~n~, and a basis of V to be these vectors in that order.There is a natural map from𝔐d+1(V~)′ to 𝔐d(V): intersecting with V an isotropic subspace of V~ of dimension d+1 gives an isotropic subspace of V of dimension d, and we denote this map by ∩. The map ∩ is onto, for every isotropic subspace of V~ (and hence of V) is contained in an isotropic subspace of V~ of dimension d+1. In fact, more is true: the map ∩ is two-to-one. The map ∩ being two-to-one, it is also elementary to see that the two points in any fiber lie one in each component of 𝔐d+1(V~)′. We therefore get a natural isomorphism between 𝔐d+1(V~) and 𝔐d(V). Therefore, now onwards we call the (even) Orthogonal Grassmannian 𝔐d(V) (as defined above for a 2d-dimensional vector space V) the Orthogonal Grassmannian.Let𝔐d(V)⊆Gd(V)↪ℙ(∧dV) be the Plücker embedding (where Gd(V) denotes the Grassmannian of all d-dimensional subspaces of V). Thus 𝔐d(V) is a closed subvariety of the projective variety Gd(V), and hence 𝔐d(V) inherits the structure of a projective variety.We takeB (resp., B-) to be the subgroup of SO(V) consisting of those elements that are upper triangular (resp., lower triangular) with respect to the basis e1,…,e2d and the subgroup T of SO(V) consisting of those elements that are diagonal with respect to e1,…,e2d. It can be easily checked that T is a maximal torus of SO(V); B and B- are the Borel subgroups of SO(V) which contain T. The group SO(V) acts transitively on 𝔐d(V), and the T-fixed points of 𝔐d(V) under this action are easily seen to be of the form 〈ei1,…,eid〉 for {i1,…,id} in I(d), where I(d) is the set of subsets of {1,…,2d} of cardinality d satisfying the following two conditions: (i)
for eachk, 1≤k≤2d, the subset contains exactly one of k, k*, and(ii)
the number of elements in the subset that exceedd is even.We writeI(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. For μ={μ1,…,μd}∈I(d,2d),μ1<⋯<μd, define the complement of μ as {1,…,2d}∖μ, and denote it by μ¯.TheB-orbits (as well as B--orbits) of 𝔐d(V) are naturally indexed by its T-fixed points: each B-orbit (as well as B--orbit) contains one and only one such point. Let α∈I(d) be arbitrary, and let eα denote the corresponding T-fixed point of 𝔐d(V). The Zariski closure of the B- (resp., B--) orbit through eα, with canonical reduced scheme structure, is called a Schubert variety (resp., opposite Schubert variety) and denoted by Xα (resp., Xα). For α,γ∈I(d), the scheme-theoretic intersection Xαγ=Xα∩Xγ is called a Richardson variety. Each B-orbit (as well as B--orbit) being irreducible and open in its closure, it follows that B-orbit closures (resp., B--orbit closures) are indexed by the B-orbits (resp., B--orbits). Thus the set I(d) becomes an indexing set for the Schubert varieties in 𝔐d(V), and the set consisting of all pairs of elements of I(d) becomes an indexing set for the Richardson varieties in 𝔐d(V). It can be shown that Xαγ is nonempty if and only if α≤γ; that for β∈I(d), eβ∈Xαγ if and only if α≤β≤γ.


## 3. Statement of the Problem and the Strategy of the Proof

### 3.1. Initial Statement of the Problem

The problem that is tackled in this paper is this: given aT-fixed point on a Richardson variety in 𝔐d(V), compute the initial ideal, with respect to some convenient term order, of the ideal of functions vanishing on the tangent cone to the Richardson variety at the given T-fixed point. The term order is specified in Section 3.4, and the answer is given in Theorem 7.For the rest of this paper,α,β, and γ are arbitrarily fixed elements of I(d) such that α≤β≤γ. So, the problem tackled in this paper can be restated as follows. Given the Richardson variety Xαγ in 𝔐d(V) and the T-fixed point eβ in it, find the initial ideal of the ideal of functions vanishing on the tangent cone at eβ to Xαγ, with respect to some conveniently chosen term order. The tangent cone being a subvariety of the tangent space at eβ to 𝔐d(V), we first choose a convenient set of coordinates for the tangent space. But for that we need to fix some notation.


### 3.2. Basic Notation

For this subsection, let us fix an arbitrary elementv of I(d,2d). 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(v) denote the set of all such ordered pairs, and set 𝔑(v):={(r,c)∈ℜ(v)|r>c}, 𝔒ℜ(v):={(r,c)∈ℜ(v)|r<c*}, 𝔒𝔑(v):={(r,c)∈ℜ(v)|r>c,r<c*}=𝔒ℜ(v)∩𝔑(v), 𝔡v:={(r,c)∈ℜ(v)|r=c*}, 𝔄ℜ(v):={(r,c)∈ℜ(v)|r>c*}, and 𝔄𝔑(v):={(r,c)∈ℜ(v)|r>c,r>c*}. We will refer to 𝔡v as the diagonal.Figure1 shows a drawing of ℜ(v). 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 and the rows from top to bottom by the entries of {1,…,2d}∖v in ascending order. The points of 𝔡v are those on the diagonal, the points of 𝔒ℜ(v) are those that are (strictly) above the diagonal, and the points of 𝔑(v) are those that are to the South-West of the polyline captioned “boundary of 𝔑(v)”—we draw the boundary so that points on the boundary belong to 𝔑(v). The reader can readily verify that d=13 and v=(1,2,3,4,6,7,10,11,13,15,18,19,and22) for the particular picture drawn. The points of 𝔒ℜ(v) indicated by solid circles form an extendedv-chain (see Figure 1); the definition of an extended v-chain is given later in Section 3.5.Figure 1Figure 2We will be consideringmonomials in some of these sets. A monomial, as usual, is a subset with each member being allowed a multiplicity (taking values in the nonnegative integers). The degree of a monomial has also the usual sense: consider the underlying set of the monomial and look at the multiplicity with which each element of this underlying set appears in the monomial; the degree of the monomial is the sum of these multiplicities. The intersection of any two monomials in a set also has the natural meaning; it is the monomial whose underlying set is the intersection of the underlying sets of the given two monomials and the multiplicity of each element being the minimum of the multiplicities with which the element occurs in the two given monomials.Given any two monomialsA and B consisting of elements of ℜ(v), let set (A) and set (B) denote the underlying sets of A and B, respectively. We say that B⊆A (as monomials) if  set (B)⊆set(A), and the multiplicity with which every element occurs in the monomial B is less than or equal to the multiplicity with which the same element occurs in the monomial A. Given two monomials A and B consisting of elements of ℜ(v) such that B⊆A (as monomials), we can define a monomial called the “monomial minus” of B from A (denoted by A∖mB) as follows. Take any element x of set(B). If the multiplicity with which x occurs in A is mx(A) and the multiplicity with which x occurs in B is mx(B), then the multiplicity with which x occurs in the monomial A∖mB is mx(A)-mx(B). And any element in set(A)∖set(B) occurs in the monomial A∖mB with the same multiplicity with which it occurs in A. This finishes the description of A∖mB.Remark 1.
Note that in this subsection, andv was any element of I(d,2d), v was not necessarily in I(d). In particular, all the above basic notations will hold true, if we take v∈I(d) as well.


### 3.3. The Ideal of the Tangent Cone toXαγ at eβ

Letβ be the element of I(d) which was fixed at the beginning of this section. Consider the matrix of size 2d×d whose columns are numbered by the entries of β and the rows by {1,…,2d}; the rows corresponding to the entries of β form the d×d identity matrix, and the remaining d rows form a skew symmetric matrix whose upper half entries are variables of the form X(r,c), where (r,c)∈𝔒ℜ(β).Let𝔐d(V)⊆Gd(V)↪ℙ(∧dV) be the Plücker embedding (where Gd(V) denotes the Grassmannian of all d-dimensional subspaces of V). For θ in I(d,2d), let pθ denote the corresponding Plücker coordinate. Consider the affine patch 𝔸 of ℙ(∧dV) given by pβ≠0. The affine patch 𝔸β:=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(β).Forθ∈I(d), consider the submatrix of the above mentioned matrix given by the rows numbered θ∖β and columns numbered β∖θ. Such a submatrix is of even size and is skew-symmetric about antidiagonal, therefore its determinant is a square. The square root, which is determined up to sign, is called the Pfaffian. Let fθ,β denote this Pfaffian. Set Yαγ(β):=Xαγ∩𝔸β. From [10] we can deduce a set of generators for the ideal I of functions on 𝔸β vanishing on Yαγ(β) (see Section 4.2 of [9] for the special case of the Schubert varieties). The following equation gives the generators:
(1)I=(fτ,β∣τ∈I(d),α≰τorτ≰γ).
We are interested in the tangent cone to Xαγ at eβ or, what is the same, the tangent cone to Yαγ(β) at the origin. Observe that fτ,β is a homogeneous polynomial. Because of this, Yαγ(β) itself is a cone and so is equal to its tangent cone at the origin. The ideal of the tangent cone to Xαγ at eβ is therefore the ideal I in (1).


### 3.4. The Term Order

We now specify the term order▹ on monomials in the coordinate functions (of the tangent space to 𝔐d(V) at the torus fixed point eβ; it is easy to see that coordinate ring of the tangent space to 𝔐d(V) at eβ is the polynomial ring in variables of the form X(r,c), where (r,c)∈𝔒ℜ(β)) with respect to which the initial ideal of the ideal I of the tangent cone is to be taken.Let> be a total order on 𝔒ℜ(β) satisfying all of the following 6 conditions.(i)
μ>ν, if μ∈𝔒𝔑(β), ν∈𝔒ℜ(β)∖𝔒𝔑(β), and the row indices of μ and ν are equal.(ii)
μ>ν, if μ∈𝔒𝔑(β), ν∈𝔒𝔑(β), the row indices of μ and ν are equal, and the column index of μ exceeds that of ν.(iii)
μ>ν if μ∈𝔒𝔑(β), ν∈𝔒ℜ(β), and the row index of μ is less than that of ν.(iv)
μ>ν, if μ∈𝔒ℜ(β)∖𝔒𝔑(β), ν∈𝔒𝔑(β), and the column indices of μ and ν are equal.(v)
μ>ν, if μ∈𝔒ℜ(β)∖𝔒𝔑(β), ν∈𝔒ℜ(β)∖𝔒𝔑(β), the column indices of μ and ν are equal, and the row index of μ exceeds that of ν.(vi)
μ>ν, if μ∈𝔒ℜ(β)∖𝔒𝔑(β), ν∈𝔒ℜ(β), and the column index of μ is less than that of ν. Note here that the first 3 conditions above are the same as the conditions put on the total order >1  as mentioned in Section 1.6 of [3]. Recall that in the paper [3], initial ideals of ideals of tangent cones at torus fixed points to the Schubert varieties in Orthogonal Grassmannians were computed, and the paper [3] did not deal with the Richardson varieties. The last 3 conditions above arise in this paper as an addition to the 3 conditions put on the total order >1 (as mentioned in Section 1.6 of [3]), because here we are dealing with the Richardson varieties in 𝔐d(V).Let▹ be the term order on monomials in 𝔒ℜ(β) given by the homogeneous lexicographic order with respect to >.Remark 2.
The total order> on 𝔒ℜ(β) satisfying the 6 properties mentioned above can be realized as a concrete total order on 𝔒ℜ(β) if we put the following extra condition on it.Let r(μ),r(ν),c(μ), and c(ν) denote the row index of μ, the row index of ν, the column index of μ, and the column index of ν, respectively. If r(μ)<r(ν), μ∈𝔒ℜ(β)∖𝔒𝔑(β), ν∈𝔒𝔑(β), and c(ν)<c(μ), then ν>μ when (r(ν),c(μ))∉𝔑(β)andμ>ν when (r(ν),c(μ))∈𝔑(β).


### 3.5. Extendedv-Chains and Associated Elements of I(d)

For this subsection, letv be an arbitrarily fixed element of I(d,2d) (not necessarily an element of I(d), unless otherwise stated). For elements λ=(R,C),μ=(r,c) of ℜ(v), we write λ>μ if R>r and C<c (Note that these are strict inequalities.). A sequence λ1>⋯>λk of elements of 𝔒ℜ(v) is called an extendedv-chain. Note that an extended v-chain can also be empty. Letting C to be an extended v-chain, we define C+:=C∩𝔒𝔑(v) and C-:=C∩(𝔒ℜ(v)∖𝔒𝔑(v)). We call C+ (resp., C-) the positive (resp., negative) parts of the extended v-chain C. We call an extended v-chain Cpositive (resp., negative) if C=C+(resp.,C=C-).The extended v-chain C is called nonempty if it has at least one element in it. Note that if we specialize to the case when v∈I(d), then whatever is called a v-chain in Section 2.4.1 of [9] is a positive extended v-chain over here. To every extended v-chain C, we will now associate 2 subsets 𝔡Cv(+) and 𝔡Cv(-) of 𝔡v (each of even cardinality), but for that we first need to fix some notation and recall certain terminology from [3, 9].Definition 3 (Pr and Pro).
Given any subsetD of 𝔒𝔑(v), let Pr(D) denote the monomial whose underlying set is given by {μ∣μ is a vertical or horizontal projection of some element of D} (both vertical and horizontal, as defined in Section 2.3 of [9]), and the cardinality of any element μ of this underlying set inside the monomial Pr(D) equals the number of elements in D whose one of the projections (vertical or horizontal) is μ. For λ=(r,c) in ℜ(v), define λ#:=(c*,r*). The involution λ↦λ# is just the reflection with respect to the diagonal 𝔡v. For a subset 𝔈 of 𝔑(v), the symbol 𝔈# has the obvious meaning. One calls 𝔈symmetric if 𝔈=𝔈#. Given any symmetric subset E of 𝔑(v), let one denote by E(top) the set E∩𝔒𝔑(v), by E(diag) the set E∩𝔡v, and by Pro(E) the monomial formed by taking the union of the subset E(diag) with the monomial Pr(E(top)). Let one make the definition of Pro(E) more precise. The multiplicity with which any element occurs in the monomial Pro(E) is equal to the sum of the multiplicities with which the element occurs in the subset E(diag) and the monomial Pr(E(top)). So for any symmetric subset E of 𝔑(v), Pro(E) is a monomial consisting of elements from the diagonal. Similarly for any subset D of 𝔒𝔑(v), Pr(D) is also a monomial consisting of elements from the diagonal.Letv∈I(d). Let C:α1=(r1,c1)>α2=(r2,c2)>⋯>αl=(rl,cl) be a positive extended v-chain in 𝔒𝔑(v). Two consecutive elements αj and αj+1 of C are said to be connected if the following conditions are both satisfied: (i)
their legs are “intertwined”, that is,rj*≥cj+1,(ii)
the point(rj+1,rj*) belongs to 𝔑(v). 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.For anyα=(r,c) in 𝔒𝔑(v), the elements pv(α):=(c*,c) and ph(α):=(r,r*) of the diagonal are called, respectively, the vertical and horizontal projections ofα. Given the positive extended v-chain C as above, let 𝔖C be the monomial of 𝔑(v) associated to C defined as follows.First suppose thatC is a connected v-chain in 𝔒𝔑(v). 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(v), then j=l. Define
(2)𝔖C:={{pv(α1),…,pv(αl)}iflis even{pv(α1),…,pv(αl)}∪{ph(αl)}iflis  odd,ph(αl)∈𝔑(v){pv(α1),…,pv(αl-1)}∪{αl,αl#}iflis  odd,ph(αl)∉𝔑(v).For av-chain C that is not necessarily connected, let C=C1∪C2∪⋯ be the partition of C into its connected components, and set
(3)𝔖C:=𝔖C1∪𝔖C2∪⋯.Note that even if we takev to be in I(d,2d) (and not merely in I(d)) and define 𝔖C for any positive extended v-chain C exactly in the same way as we did above, there is no logical inconsistency. Hence we extend the definition of 𝔖C to any positive extended v-chain C, where v∈I(d,2d). Clearly 𝔖C is a symmetric subset of 𝔑(v). Hence the monomial Pro(𝔖C) is well defined for any positive extended v-chain C, where v∈I(d,2d).Definition 4 (the flip mapF).
For anyv∈I(d,2d) and any element λ=(r,c)∈ℜ(v), let F(λ) be the element of ℜ(v*) given by F(λ):=(c,r). So F is an invertible map from ℜ(v) to ℜ(v*) (note here that if v∈I(d), then v* needs not always to belong to I(d)); let one denote the inverse of F by F-1. The map F naturally induces an invertible map from the set of all monomials in ℜ(v) to the set of all monomials in ℜ(v*). One continues to call the induced map F and its inverse F-1.Definition 5 (the subsets𝔡Cv(+) and 𝔡Cv(-) of 𝔡v).
Given any non-empty extendedv-chain C, one will now associate 2 subsets 𝔡Cv(+) and 𝔡Cv(-) of 𝔡v (each of even cardinality) with it. Let
(4)𝔡Cv(+):={F-1(Pr(F(C))∖mPro(𝔖F(C)))ifCis  negativePro(𝔖C+)if  otherwise.
Similarly, let
(5)𝔡Cv(-):={Pr(C)∖mPro(𝔖C)ifCis positiveF-1(Pro(𝔖F(C-)))if  otherwise.
It is an easy exercise to check that 𝔡Cv(+) and 𝔡Cv(-) thus defined are actually subsets (not monomials) of 𝔡v and that each of them has even cardinality.Definition 6 (elements ofI(d)  associated  with𝔡Cv(+)  and  𝔡Cv(-)).
For this definition, we letv to be an arbitrary element in I(d). Note that given any subset S of 𝔡v of even cardinality, one can naturally associate an element of I(d) to it by removing those entries from v which appear as column indices in the elements of the set S and then adding to it the row indices of all the elements of S. It is easy to check that the resulting element actually belongs to I(d). One denotes the resulting element by I(d)(S). If S is empty, then I(d)(S) is taken to be v itself.
LetwC+(v):=I(d)(𝔡Cv(+)) and wC-(v):=I(d)(𝔡Cv(-)). These are the two elements of I(d) that we can naturally associate with the subsets 𝔡Cv(+) and 𝔡Cv(-) of 𝔡v.


### 3.6. The Main Theorem and a Strategy of the Proof

Recall that the ideal of the tangent cone toXαγ at eβ is the ideal I given by (1). Let ▹ be as in Section 3.4. For any element f∈I; let in▹f denote the initial term of f with respect to the term order ▹. We define in▹I to be the ideal 〈in▹f|f∈I〉 inside the polynomial ring P:=𝔨[X(r,c)∣(r,c)∈𝔒ℜ(β)]. For any monomial U in 𝔒ℜ(β), let us denote by XU the product of all the elements X(r,c), where (r,c) runs over the elements in U with multiplicities.LetChainsαγ(β) denote the set {XC|Cis a nonempty extendedβ-chain in𝔒ℜ(β) such that either (i) or (ii) below holds}.(i)C- is nonempty, and α≰wC--(β).  (ii) C+ is nonempty, and wC++(β)≰γ.The main result of this paper is the following.Theorem 7.
in
▹
I
=
〈
Chains
α
γ
(
β
)
〉.Example 8.
Letd=5, α=(1,2,3,6,7),β=(1,2,4,6,8), and γ=(1,2,5,7,8). Clearly then α≤β≤γ. It can be verified that the set Chainsαγ(β) consists of elements of the following 3 types.(i)
Any extendedβ-chain in 𝔒ℜ(β) which contains the element (3,6).(ii)
Any extendedβ-chain in 𝔒ℜ(β) which contains the element (3,4) and a nonempty positive part.(iii)
All positive extendedβ-chain in 𝔒ℜ(β)   except the singleton-β-chain {(5,4)}.
One can therefore determine the corresponding initial idealin▹I using Theorem 7.Remark 9.
It follows from the statement of Theorem7 that the set of all monomials in 𝔒ℜ(β) which contain at least one extended β-chain C such that XC∈Chainsαγ(β) form a vector space basis of the initial ideal in▹I over the field 𝔨; see, for example, Example 8. In the special case when the Richardson variety is a Schubert variety, it is easy to see that the previous statement of this remark says exactly what has been said in the main theorem (Theorem 1.8.1) of [3].Remark 10.
Sincein▹I=〈Chainsαγ(β)〉, it follows that the dimension as a vector space of the graded piece of P/in▹I of degree m equals the cardinality of the set of all monomials in 𝔒ℜ(β) of degree m which do not contain any extended β-chain C such that XC∈Chainsαγ(β).
SinceP/I and P/in▹I have the same Hilbert function, the same is true with P/in▹I (in the previous statement) replaced by P/I. Hence if Rαγ(β) denotes the coordinate ring of the tangent cone to Xαγ at eβ, then the dimension as a vector space of the homogeneous piece of Rαγ(β) of degree mequals the cardinality of the set of all monomials in 𝔒ℜ(β) of degree m which are such that for every extended β-chain C in the monomial, α≤wC--(β) and wC++(β)≤γ.From Remark10, the following corollary is immediate (the proof is similar to that of Corollary 2.5.2 of [9]).Corollary 11.
The multiplicity ofXαγ at eβ equals the number of monomials in 𝔒ℜ(β) of maximal cardinality that are square-free and are such that, given any extended β-chain C in any of these monomials, one has: α≤wC--(β) and wC++(β)≤γ.We now briefly sketch the proof of Theorem7 (omitting details, which can be found in Section 8). In order to introduce the main combinatorial objects of interest and outline a strategy of the proof, we will first need to prove that the set Chainsαγ(β)⊆in▹I, and this proof will follow from whatever is said in Remark 12.Remark 12.
LetC be a nonempty extended β-chain in 𝔒ℜ(β) such that XC∈Chainsαγ(β). If C+ is nonempty and wC++(β)≰γ, then it can be proved that XC+∈in▹I, the proof being exactly the same as that in Section 4 of [3]. Then since in▹I is an ideal and XC=XC-XC+, it follows that XC∈in▹I.
IfC- is nonempty and α≰wC--(β), look at F(C-), where F is the flip map as defined in Definition 4 from the set of all monomials in ℜ(β) to the set of all monomials in ℜ(β*). Then F(C-) is a positive extended β*-chain in 𝔒ℜ(β*). We need to prove that XC∈in▹I, for which it is enough to show that XC-∈in▹I. To prove that XC-∈in▹I, we will proceed in a way equivalent to the proof done in Section 4 of [3]. But there is a subtle difference between what is proved in Section 4 of [3] and what we are going to prove here; namely, Whatever was proved in Section 4 of [3] can be rephrased in the language of this paper as “Every positive extended β-chain D satisfying the property that wD+(β)≰γ belongs to the initial ideal of the ideal of the tangent cone,” but here we are going to prove that “Every negative extended β-chain D satisfying the property that α≰wD-(β) belongs to the initial ideal of the ideal of the tangent cone.”
Because of this subtle difference, we need to construct certain gadgets for negative extendedβ-chains, which will play a role similar to the role of the objects like the new forms, Proj and Proje corresponding to positive extended β-chains (For positive extended β-chains, such objects are already defined in [3]). This construction is given in the following paragraph.
Consider the positive extendedβ*-chain F(C-). We can construct new forms, Proj and Proje for F(C-) in the same way as they were constructed in [3], note here the fact that β* may or may not belong to I(d) does not really affect the construction of the new forms, Proj and Proje for F(C-). Then we apply the map F-1 to these objects constructed for F(C-), the resulting objects are the analogues of the new forms, Proj and Proje for the negative extended β-chain C-. We apply a similar treatment to any other monomial related to F(C-) that we happen to encounter if we replace the “v-chain A” in Section 4.2 of [3] with “the positive extended β*-chain F(C-).”
In Section 2.4 of [3], an element yE of I(d) corresponding to any v-chain E (the notion of a v-chain being as in Section 1.7 of [3]) has been defined. The analogous element of I(d) for the negative extended β-chain C- (we call it yC- here) can be obtained from F-1(Proje(F(C-))) by following the natural process: the column indices of elements ofF-1(Proje(F(C-)))occur as members ofβ; these are replaced by the row indices to obtain the desired element ofI(d)forC-. It is easy to check that yC- belongs to I(d) and that yC-≤wC--(β)≤β. Since we already have that α≰wC--(β), it follows that α≰yC-. These facts about yC- will be needed to produce an analogue of the main proof of [3] in our present case. To be more precise, these facts about yC- give the analogues of Propositions 2.4.1 and 2.4.2 of [3], and these two propositions had been used quite crucially inside the main proof of [3].
With all these analogues constructed for negative extendedβ-chains, we can proceed in an “equivalent” manner (Here, by “equivalent” we mean keeping track of the subtle difference as mentioned above and working accordingly.) as in the paper [3] and end up proving the desired fact, namely, XC-∈in▹I.SinceChainsαγ(β)⊆in▹I, we have 〈Chainsαγ(β)〉⊆in▹I. To prove Theorem 7, we now need to show that 〈Chainsαγ(β)〉⊇in▹I. For this, it suffices to show that, in any degree, the number of monomials of 〈Chainsαγ(β)〉 is ≥ the number of monomials of in▹I. Or equivalently, it suffices to show that, in any degree, the number of monomials of P/〈Chainsαγ(β)〉 is ≤ the number of monomials of P/in▹I.Consider the affine patchYαγ(β)(:=Xαγ∩𝔸β) of the Richardson variety Xαγ. The following is a well known result.Theorem 13.
𝔨
[
Y
α
γ
(
β
)
]
=
P
/
I, where P=𝔨[X(r,c)|(r,c)∈𝔒ℜ(β)] and I is as in (1).Both the monomials ofP/in▹I and the standard monomials on Yαγ(β) form a basis for P/I, and thus agree in cardinality in any degree. Therefore, to prove, that in any degree, the number of monomials of P/〈Chainsαγ(β)〉 is ≤ the number of monomials of P/in▹I, it suffices to give a degree-preserving injection from the set of all monomials in P/〈Chainsαγ(β)〉 to the set of all standard monomials on Yαγ(β). We construct such an injection, the Orthogonal-bounded-RSK (OBRSK), from an indexing set of the former to an indexing set of the later. These indexing sets are given in Table 1.Table 1
Two subsets of the ringP=𝔨[X(r,c)|(r,c)∈𝔒ℜ(β)] and their indexing sets.
Set of elements inP=𝔨[X(r,c)|(r,c)∈𝔒ℜ(β)]
Indexing set
Monomials ofP/〈Chainsαγ(β)〉
Pairs of nonempty skew-symmetric lexicographic multisets onβ-×β bounded by Tα,Wγ
Standard monomials onYαγ(β)
Nonvanishing skew-symmetric notched bitableaux onβ-×β bounded by Tα,Wγ


## 3.1. Initial Statement of the Problem

The problem that is tackled in this paper is this: given aT-fixed point on a Richardson variety in 𝔐d(V), compute the initial ideal, with respect to some convenient term order, of the ideal of functions vanishing on the tangent cone to the Richardson variety at the given T-fixed point. The term order is specified in Section 3.4, and the answer is given in Theorem 7.For the rest of this paper,α,β, and γ are arbitrarily fixed elements of I(d) such that α≤β≤γ. So, the problem tackled in this paper can be restated as follows. Given the Richardson variety Xαγ in 𝔐d(V) and the T-fixed point eβ in it, find the initial ideal of the ideal of functions vanishing on the tangent cone at eβ to Xαγ, with respect to some conveniently chosen term order. The tangent cone being a subvariety of the tangent space at eβ to 𝔐d(V), we first choose a convenient set of coordinates for the tangent space. But for that we need to fix some notation.


## 3.2. Basic Notation

For this subsection, let us fix an arbitrary elementv of I(d,2d). 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(v) denote the set of all such ordered pairs, and set 𝔑(v):={(r,c)∈ℜ(v)|r>c}, 𝔒ℜ(v):={(r,c)∈ℜ(v)|r<c*}, 𝔒𝔑(v):={(r,c)∈ℜ(v)|r>c,r<c*}=𝔒ℜ(v)∩𝔑(v), 𝔡v:={(r,c)∈ℜ(v)|r=c*}, 𝔄ℜ(v):={(r,c)∈ℜ(v)|r>c*}, and 𝔄𝔑(v):={(r,c)∈ℜ(v)|r>c,r>c*}. We will refer to 𝔡v as the diagonal.Figure1 shows a drawing of ℜ(v). 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 and the rows from top to bottom by the entries of {1,…,2d}∖v in ascending order. The points of 𝔡v are those on the diagonal, the points of 𝔒ℜ(v) are those that are (strictly) above the diagonal, and the points of 𝔑(v) are those that are to the South-West of the polyline captioned “boundary of 𝔑(v)”—we draw the boundary so that points on the boundary belong to 𝔑(v). The reader can readily verify that d=13 and v=(1,2,3,4,6,7,10,11,13,15,18,19,and22) for the particular picture drawn. The points of 𝔒ℜ(v) indicated by solid circles form an extendedv-chain (see Figure 1); the definition of an extended v-chain is given later in Section 3.5.Figure 1Figure 2We will be consideringmonomials in some of these sets. A monomial, as usual, is a subset with each member being allowed a multiplicity (taking values in the nonnegative integers). The degree of a monomial has also the usual sense: consider the underlying set of the monomial and look at the multiplicity with which each element of this underlying set appears in the monomial; the degree of the monomial is the sum of these multiplicities. The intersection of any two monomials in a set also has the natural meaning; it is the monomial whose underlying set is the intersection of the underlying sets of the given two monomials and the multiplicity of each element being the minimum of the multiplicities with which the element occurs in the two given monomials.Given any two monomialsA and B consisting of elements of ℜ(v), let set (A) and set (B) denote the underlying sets of A and B, respectively. We say that B⊆A (as monomials) if  set (B)⊆set(A), and the multiplicity with which every element occurs in the monomial B is less than or equal to the multiplicity with which the same element occurs in the monomial A. Given two monomials A and B consisting of elements of ℜ(v) such that B⊆A (as monomials), we can define a monomial called the “monomial minus” of B from A (denoted by A∖mB) as follows. Take any element x of set(B). If the multiplicity with which x occurs in A is mx(A) and the multiplicity with which x occurs in B is mx(B), then the multiplicity with which x occurs in the monomial A∖mB is mx(A)-mx(B). And any element in set(A)∖set(B) occurs in the monomial A∖mB with the same multiplicity with which it occurs in A. This finishes the description of A∖mB.Remark 1.
Note that in this subsection, andv was any element of I(d,2d), v was not necessarily in I(d). In particular, all the above basic notations will hold true, if we take v∈I(d) as well.


## 3.3. The Ideal of the Tangent Cone toXαγ at eβ

Letβ be the element of I(d) which was fixed at the beginning of this section. Consider the matrix of size 2d×d whose columns are numbered by the entries of β and the rows by {1,…,2d}; the rows corresponding to the entries of β form the d×d identity matrix, and the remaining d rows form a skew symmetric matrix whose upper half entries are variables of the form X(r,c), where (r,c)∈𝔒ℜ(β).Let𝔐d(V)⊆Gd(V)↪ℙ(∧dV) be the Plücker embedding (where Gd(V) denotes the Grassmannian of all d-dimensional subspaces of V). For θ in I(d,2d), let pθ denote the corresponding Plücker coordinate. Consider the affine patch 𝔸 of ℙ(∧dV) given by pβ≠0. The affine patch 𝔸β:=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(β).Forθ∈I(d), consider the submatrix of the above mentioned matrix given by the rows numbered θ∖β and columns numbered β∖θ. Such a submatrix is of even size and is skew-symmetric about antidiagonal, therefore its determinant is a square. The square root, which is determined up to sign, is called the Pfaffian. Let fθ,β denote this Pfaffian. Set Yαγ(β):=Xαγ∩𝔸β. From [10] we can deduce a set of generators for the ideal I of functions on 𝔸β vanishing on Yαγ(β) (see Section 4.2 of [9] for the special case of the Schubert varieties). The following equation gives the generators:
(1)I=(fτ,β∣τ∈I(d),α≰τorτ≰γ).
We are interested in the tangent cone to Xαγ at eβ or, what is the same, the tangent cone to Yαγ(β) at the origin. Observe that fτ,β is a homogeneous polynomial. Because of this, Yαγ(β) itself is a cone and so is equal to its tangent cone at the origin. The ideal of the tangent cone to Xαγ at eβ is therefore the ideal I in (1).


## 3.4. The Term Order

We now specify the term order▹ on monomials in the coordinate functions (of the tangent space to 𝔐d(V) at the torus fixed point eβ; it is easy to see that coordinate ring of the tangent space to 𝔐d(V) at eβ is the polynomial ring in variables of the form X(r,c), where (r,c)∈𝔒ℜ(β)) with respect to which the initial ideal of the ideal I of the tangent cone is to be taken.Let> be a total order on 𝔒ℜ(β) satisfying all of the following 6 conditions.(i)
μ>ν, if μ∈𝔒𝔑(β), ν∈𝔒ℜ(β)∖𝔒𝔑(β), and the row indices of μ and ν are equal.(ii)
μ>ν, if μ∈𝔒𝔑(β), ν∈𝔒𝔑(β), the row indices of μ and ν are equal, and the column index of μ exceeds that of ν.(iii)
μ>ν if μ∈𝔒𝔑(β), ν∈𝔒ℜ(β), and the row index of μ is less than that of ν.(iv)
μ>ν, if μ∈𝔒ℜ(β)∖𝔒𝔑(β), ν∈𝔒𝔑(β), and the column indices of μ and ν are equal.(v)
μ>ν, if μ∈𝔒ℜ(β)∖𝔒𝔑(β), ν∈𝔒ℜ(β)∖𝔒𝔑(β), the column indices of μ and ν are equal, and the row index of μ exceeds that of ν.(vi)
μ>ν, if μ∈𝔒ℜ(β)∖𝔒𝔑(β), ν∈𝔒ℜ(β), and the column index of μ is less than that of ν. Note here that the first 3 conditions above are the same as the conditions put on the total order >1  as mentioned in Section 1.6 of [3]. Recall that in the paper [3], initial ideals of ideals of tangent cones at torus fixed points to the Schubert varieties in Orthogonal Grassmannians were computed, and the paper [3] did not deal with the Richardson varieties. The last 3 conditions above arise in this paper as an addition to the 3 conditions put on the total order >1 (as mentioned in Section 1.6 of [3]), because here we are dealing with the Richardson varieties in 𝔐d(V).Let▹ be the term order on monomials in 𝔒ℜ(β) given by the homogeneous lexicographic order with respect to >.Remark 2.
The total order> on 𝔒ℜ(β) satisfying the 6 properties mentioned above can be realized as a concrete total order on 𝔒ℜ(β) if we put the following extra condition on it.Let r(μ),r(ν),c(μ), and c(ν) denote the row index of μ, the row index of ν, the column index of μ, and the column index of ν, respectively. If r(μ)<r(ν), μ∈𝔒ℜ(β)∖𝔒𝔑(β), ν∈𝔒𝔑(β), and c(ν)<c(μ), then ν>μ when (r(ν),c(μ))∉𝔑(β)andμ>ν when (r(ν),c(μ))∈𝔑(β).


## 3.5. Extendedv-Chains and Associated Elements of I(d)

For this subsection, letv be an arbitrarily fixed element of I(d,2d) (not necessarily an element of I(d), unless otherwise stated). For elements λ=(R,C),μ=(r,c) of ℜ(v), we write λ>μ if R>r and C<c (Note that these are strict inequalities.). A sequence λ1>⋯>λk of elements of 𝔒ℜ(v) is called an extendedv-chain. Note that an extended v-chain can also be empty. Letting C to be an extended v-chain, we define C+:=C∩𝔒𝔑(v) and C-:=C∩(𝔒ℜ(v)∖𝔒𝔑(v)). We call C+ (resp., C-) the positive (resp., negative) parts of the extended v-chain C. We call an extended v-chain Cpositive (resp., negative) if C=C+(resp.,C=C-).The extended v-chain C is called nonempty if it has at least one element in it. Note that if we specialize to the case when v∈I(d), then whatever is called a v-chain in Section 2.4.1 of [9] is a positive extended v-chain over here. To every extended v-chain C, we will now associate 2 subsets 𝔡Cv(+) and 𝔡Cv(-) of 𝔡v (each of even cardinality), but for that we first need to fix some notation and recall certain terminology from [3, 9].Definition 3 (Pr and Pro).
Given any subsetD of 𝔒𝔑(v), let Pr(D) denote the monomial whose underlying set is given by {μ∣μ is a vertical or horizontal projection of some element of D} (both vertical and horizontal, as defined in Section 2.3 of [9]), and the cardinality of any element μ of this underlying set inside the monomial Pr(D) equals the number of elements in D whose one of the projections (vertical or horizontal) is μ. For λ=(r,c) in ℜ(v), define λ#:=(c*,r*). The involution λ↦λ# is just the reflection with respect to the diagonal 𝔡v. For a subset 𝔈 of 𝔑(v), the symbol 𝔈# has the obvious meaning. One calls 𝔈symmetric if 𝔈=𝔈#. Given any symmetric subset E of 𝔑(v), let one denote by E(top) the set E∩𝔒𝔑(v), by E(diag) the set E∩𝔡v, and by Pro(E) the monomial formed by taking the union of the subset E(diag) with the monomial Pr(E(top)). Let one make the definition of Pro(E) more precise. The multiplicity with which any element occurs in the monomial Pro(E) is equal to the sum of the multiplicities with which the element occurs in the subset E(diag) and the monomial Pr(E(top)). So for any symmetric subset E of 𝔑(v), Pro(E) is a monomial consisting of elements from the diagonal. Similarly for any subset D of 𝔒𝔑(v), Pr(D) is also a monomial consisting of elements from the diagonal.Letv∈I(d). Let C:α1=(r1,c1)>α2=(r2,c2)>⋯>αl=(rl,cl) be a positive extended v-chain in 𝔒𝔑(v). Two consecutive elements αj and αj+1 of C are said to be connected if the following conditions are both satisfied: (i)
their legs are “intertwined”, that is,rj*≥cj+1,(ii)
the point(rj+1,rj*) belongs to 𝔑(v). 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.For anyα=(r,c) in 𝔒𝔑(v), the elements pv(α):=(c*,c) and ph(α):=(r,r*) of the diagonal are called, respectively, the vertical and horizontal projections ofα. Given the positive extended v-chain C as above, let 𝔖C be the monomial of 𝔑(v) associated to C defined as follows.First suppose thatC is a connected v-chain in 𝔒𝔑(v). 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(v), then j=l. Define
(2)𝔖C:={{pv(α1),…,pv(αl)}iflis even{pv(α1),…,pv(αl)}∪{ph(αl)}iflis  odd,ph(αl)∈𝔑(v){pv(α1),…,pv(αl-1)}∪{αl,αl#}iflis  odd,ph(αl)∉𝔑(v).For av-chain C that is not necessarily connected, let C=C1∪C2∪⋯ be the partition of C into its connected components, and set
(3)𝔖C:=𝔖C1∪𝔖C2∪⋯.Note that even if we takev to be in I(d,2d) (and not merely in I(d)) and define 𝔖C for any positive extended v-chain C exactly in the same way as we did above, there is no logical inconsistency. Hence we extend the definition of 𝔖C to any positive extended v-chain C, where v∈I(d,2d). Clearly 𝔖C is a symmetric subset of 𝔑(v). Hence the monomial Pro(𝔖C) is well defined for any positive extended v-chain C, where v∈I(d,2d).Definition 4 (the flip mapF).
For anyv∈I(d,2d) and any element λ=(r,c)∈ℜ(v), let F(λ) be the element of ℜ(v*) given by F(λ):=(c,r). So F is an invertible map from ℜ(v) to ℜ(v*) (note here that if v∈I(d), then v* needs not always to belong to I(d)); let one denote the inverse of F by F-1. The map F naturally induces an invertible map from the set of all monomials in ℜ(v) to the set of all monomials in ℜ(v*). One continues to call the induced map F and its inverse F-1.Definition 5 (the subsets𝔡Cv(+) and 𝔡Cv(-) of 𝔡v).
Given any non-empty extendedv-chain C, one will now associate 2 subsets 𝔡Cv(+) and 𝔡Cv(-) of 𝔡v (each of even cardinality) with it. Let
(4)𝔡Cv(+):={F-1(Pr(F(C))∖mPro(𝔖F(C)))ifCis  negativePro(𝔖C+)if  otherwise.
Similarly, let
(5)𝔡Cv(-):={Pr(C)∖mPro(𝔖C)ifCis positiveF-1(Pro(𝔖F(C-)))if  otherwise.
It is an easy exercise to check that 𝔡Cv(+) and 𝔡Cv(-) thus defined are actually subsets (not monomials) of 𝔡v and that each of them has even cardinality.Definition 6 (elements ofI(d)  associated  with𝔡Cv(+)  and  𝔡Cv(-)).
For this definition, we letv to be an arbitrary element in I(d). Note that given any subset S of 𝔡v of even cardinality, one can naturally associate an element of I(d) to it by removing those entries from v which appear as column indices in the elements of the set S and then adding to it the row indices of all the elements of S. It is easy to check that the resulting element actually belongs to I(d). One denotes the resulting element by I(d)(S). If S is empty, then I(d)(S) is taken to be v itself.
LetwC+(v):=I(d)(𝔡Cv(+)) and wC-(v):=I(d)(𝔡Cv(-)). These are the two elements of I(d) that we can naturally associate with the subsets 𝔡Cv(+) and 𝔡Cv(-) of 𝔡v.


## 3.6. The Main Theorem and a Strategy of the Proof

Recall that the ideal of the tangent cone toXαγ at eβ is the ideal I given by (1). Let ▹ be as in Section 3.4. For any element f∈I; let in▹f denote the initial term of f with respect to the term order ▹. We define in▹I to be the ideal 〈in▹f|f∈I〉 inside the polynomial ring P:=𝔨[X(r,c)∣(r,c)∈𝔒ℜ(β)]. For any monomial U in 𝔒ℜ(β), let us denote by XU the product of all the elements X(r,c), where (r,c) runs over the elements in U with multiplicities.LetChainsαγ(β) denote the set {XC|Cis a nonempty extendedβ-chain in𝔒ℜ(β) such that either (i) or (ii) below holds}.(i)C- is nonempty, and α≰wC--(β).  (ii) C+ is nonempty, and wC++(β)≰γ.The main result of this paper is the following.Theorem 7.
in
▹
I
=
〈
Chains
α
γ
(
β
)
〉.Example 8.
Letd=5, α=(1,2,3,6,7),β=(1,2,4,6,8), and γ=(1,2,5,7,8). Clearly then α≤β≤γ. It can be verified that the set Chainsαγ(β) consists of elements of the following 3 types.(i)
Any extendedβ-chain in 𝔒ℜ(β) which contains the element (3,6).(ii)
Any extendedβ-chain in 𝔒ℜ(β) which contains the element (3,4) and a nonempty positive part.(iii)
All positive extendedβ-chain in 𝔒ℜ(β)   except the singleton-β-chain {(5,4)}.
One can therefore determine the corresponding initial idealin▹I using Theorem 7.Remark 9.
It follows from the statement of Theorem7 that the set of all monomials in 𝔒ℜ(β) which contain at least one extended β-chain C such that XC∈Chainsαγ(β) form a vector space basis of the initial ideal in▹I over the field 𝔨; see, for example, Example 8. In the special case when the Richardson variety is a Schubert variety, it is easy to see that the previous statement of this remark says exactly what has been said in the main theorem (Theorem 1.8.1) of [3].Remark 10.
Sincein▹I=〈Chainsαγ(β)〉, it follows that the dimension as a vector space of the graded piece of P/in▹I of degree m equals the cardinality of the set of all monomials in 𝔒ℜ(β) of degree m which do not contain any extended β-chain C such that XC∈Chainsαγ(β).
SinceP/I and P/in▹I have the same Hilbert function, the same is true with P/in▹I (in the previous statement) replaced by P/I. Hence if Rαγ(β) denotes the coordinate ring of the tangent cone to Xαγ at eβ, then the dimension as a vector space of the homogeneous piece of Rαγ(β) of degree mequals the cardinality of the set of all monomials in 𝔒ℜ(β) of degree m which are such that for every extended β-chain C in the monomial, α≤wC--(β) and wC++(β)≤γ.From Remark10, the following corollary is immediate (the proof is similar to that of Corollary 2.5.2 of [9]).Corollary 11.
The multiplicity ofXαγ at eβ equals the number of monomials in 𝔒ℜ(β) of maximal cardinality that are square-free and are such that, given any extended β-chain C in any of these monomials, one has: α≤wC--(β) and wC++(β)≤γ.We now briefly sketch the proof of Theorem7 (omitting details, which can be found in Section 8). In order to introduce the main combinatorial objects of interest and outline a strategy of the proof, we will first need to prove that the set Chainsαγ(β)⊆in▹I, and this proof will follow from whatever is said in Remark 12.Remark 12.
LetC be a nonempty extended β-chain in 𝔒ℜ(β) such that XC∈Chainsαγ(β). If C+ is nonempty and wC++(β)≰γ, then it can be proved that XC+∈in▹I, the proof being exactly the same as that in Section 4 of [3]. Then since in▹I is an ideal and XC=XC-XC+, it follows that XC∈in▹I.
IfC- is nonempty and α≰wC--(β), look at F(C-), where F is the flip map as defined in Definition 4 from the set of all monomials in ℜ(β) to the set of all monomials in ℜ(β*). Then F(C-) is a positive extended β*-chain in 𝔒ℜ(β*). We need to prove that XC∈in▹I, for which it is enough to show that XC-∈in▹I. To prove that XC-∈in▹I, we will proceed in a way equivalent to the proof done in Section 4 of [3]. But there is a subtle difference between what is proved in Section 4 of [3] and what we are going to prove here; namely, Whatever was proved in Section 4 of [3] can be rephrased in the language of this paper as “Every positive extended β-chain D satisfying the property that wD+(β)≰γ belongs to the initial ideal of the ideal of the tangent cone,” but here we are going to prove that “Every negative extended β-chain D satisfying the property that α≰wD-(β) belongs to the initial ideal of the ideal of the tangent cone.”
Because of this subtle difference, we need to construct certain gadgets for negative extendedβ-chains, which will play a role similar to the role of the objects like the new forms, Proj and Proje corresponding to positive extended β-chains (For positive extended β-chains, such objects are already defined in [3]). This construction is given in the following paragraph.
Consider the positive extendedβ*-chain F(C-). We can construct new forms, Proj and Proje for F(C-) in the same way as they were constructed in [3], note here the fact that β* may or may not belong to I(d) does not really affect the construction of the new forms, Proj and Proje for F(C-). Then we apply the map F-1 to these objects constructed for F(C-), the resulting objects are the analogues of the new forms, Proj and Proje for the negative extended β-chain C-. We apply a similar treatment to any other monomial related to F(C-) that we happen to encounter if we replace the “v-chain A” in Section 4.2 of [3] with “the positive extended β*-chain F(C-).”
In Section 2.4 of [3], an element yE of I(d) corresponding to any v-chain E (the notion of a v-chain being as in Section 1.7 of [3]) has been defined. The analogous element of I(d) for the negative extended β-chain C- (we call it yC- here) can be obtained from F-1(Proje(F(C-))) by following the natural process: the column indices of elements ofF-1(Proje(F(C-)))occur as members ofβ; these are replaced by the row indices to obtain the desired element ofI(d)forC-. It is easy to check that yC- belongs to I(d) and that yC-≤wC--(β)≤β. Since we already have that α≰wC--(β), it follows that α≰yC-. These facts about yC- will be needed to produce an analogue of the main proof of [3] in our present case. To be more precise, these facts about yC- give the analogues of Propositions 2.4.1 and 2.4.2 of [3], and these two propositions had been used quite crucially inside the main proof of [3].
With all these analogues constructed for negative extendedβ-chains, we can proceed in an “equivalent” manner (Here, by “equivalent” we mean keeping track of the subtle difference as mentioned above and working accordingly.) as in the paper [3] and end up proving the desired fact, namely, XC-∈in▹I.SinceChainsαγ(β)⊆in▹I, we have 〈Chainsαγ(β)〉⊆in▹I. To prove Theorem 7, we now need to show that 〈Chainsαγ(β)〉⊇in▹I. For this, it suffices to show that, in any degree, the number of monomials of 〈Chainsαγ(β)〉 is ≥ the number of monomials of in▹I. Or equivalently, it suffices to show that, in any degree, the number of monomials of P/〈Chainsαγ(β)〉 is ≤ the number of monomials of P/in▹I.Consider the affine patchYαγ(β)(:=Xαγ∩𝔸β) of the Richardson variety Xαγ. The following is a well known result.Theorem 13.
𝔨
[
Y
α
γ
(
β
)
]
=
P
/
I, where P=𝔨[X(r,c)|(r,c)∈𝔒ℜ(β)] and I is as in (1).Both the monomials ofP/in▹I and the standard monomials on Yαγ(β) form a basis for P/I, and thus agree in cardinality in any degree. Therefore, to prove, that in any degree, the number of monomials of P/〈Chainsαγ(β)〉 is ≤ the number of monomials of P/in▹I, it suffices to give a degree-preserving injection from the set of all monomials in P/〈Chainsαγ(β)〉 to the set of all standard monomials on Yαγ(β). We construct such an injection, the Orthogonal-bounded-RSK (OBRSK), from an indexing set of the former to an indexing set of the later. These indexing sets are given in Table 1.Table 1
Two subsets of the ringP=𝔨[X(r,c)|(r,c)∈𝔒ℜ(β)] and their indexing sets.
Set of elements inP=𝔨[X(r,c)|(r,c)∈𝔒ℜ(β)]
Indexing set
Monomials ofP/〈Chainsαγ(β)〉
Pairs of nonempty skew-symmetric lexicographic multisets onβ-×β bounded by Tα,Wγ
Standard monomials onYαγ(β)
Nonvanishing skew-symmetric notched bitableaux onβ-×β bounded by Tα,Wγ


## 4. Skew-Symmetric Notched Bitableaux

In this section onwards, the terminology and notation of Sections 4 and 5 of [8] will be in force. Recall the definition of a semistandard notched bitableau from Section 5 of [8].Definition 14 (dual of an element with respect to a semistandard notched bitableau).
Let(P,Q) be a semistandard notched bitableau. Let pi,j (resp., qi,j) denote the entry in the ith row and jth column of P (resp., Q). For any row number i of P (or of Q), let ki denote the total number of entries in the ith row of P (or Q). One calls the entry qi,ki+1-j of Q the dual of the entry pi,j of P  with respect to  (P,Q). Similarly, we call the entry pi,ki+1-j of P the dual of the entry qi,j of Q  with respect to  (P,Q).Note that any entry ofP or Q can be identified uniquely by specifying 4 coordinates, namely, the entry x, the tableau A (A=P or Q) in which the entry lies, the row number i of the entry in the tableau A, and the column number j of the entry in the tableau A. Let set(P,Q) denote the set of all 4-tuples of the form (x,A,i,j). Given any 4-tuple (x,A,i,j)∈set(P,Q), let us denote by D(P,Q)(x,A,i,j) the 4-tuple which represents the dual of x with respect to (P,Q) (as defined in Definition 14). For (x,A,i,j),(x′,A′,i′,j′)∈set(P,Q), we say that (x,A,i,j)⪯(x′,A′,i′,j′) if x≤x′. Similarly, we say that (x,A,i,j)≺(x′,A′,i′,j′) if x<x′ and (x,A,i,j)≃(x′,A′,i′,j′) if x=x′.A semistandard notched bitableau(P,Q) is said to be skew-symmetric if the following 2 conditions are satisfied simultaneously.(i)
The bitableau(P,Q) should be of even size; that is, the number of elements in each row of P and Q should be even.(ii)
For any(x,A,i,j),(x′,A′,i′,j′)∈set(P,Q), (x,A,i,j)⪯(x′,A′,i′,j′)   if and only ifD(P,Q)(x,A,i,j)≽D(P,Q)(x′,A′,i′,j′). Moreover, (x,A,i,j)≺(x′,A′,i′,j′)  if and only ifD(P,Q)(x,A,i,j)≻D(P,Q)(x′,A′,i′,j′), and (x,A,i,j)≃(x′,A′,i′,j′)   if and only ifD(P,Q)(x,A,i,j)≃D(P,Q)(x′,A′,i′,j′).Property (ii) will be henceforth referred to asthe duality property associated to the skew-symmetric notched bitableau (P,Q). Note that a skew-symmetric notched bitableau is a semistandard notched bitableau by default. The degree of a skew-symmetric notched bitableau (P,Q) is the total number of boxes in P (or Q). The notions of negative, positive, and nonvanishing skew-symmetric notched bitableau remain the same as in Section 5 of [8]. The notion of a skew-symmetric notched bitableau (P,Q) being bounded byT,W (where T,W are subsets of ℕ2), and the notion of negative and positive parts of a skew-symmetric notched bitableau (P,Q) remain the same as they were in Section 5 of [8].If(P,Q) is a nonvanishing skew-symmetric notched bitableau, define ι(P,Q) to be the notched bitableau obtained by reversing the order of the rows of (Q,P). One checks that ι(P,Q) is a nonvanishing skew-symmetric notched bitableau. The map ι is an involution, and it maps negative skew-symmetric notched bitableaux to positive ones and vice versa. Thus ι gives a bijective pairing between the sets of negative and positive skew-symmetric notched bitableaux.


## 5. Pairs of Skew-Symmetric Lexicographic Multisets onℕ2

Recall the notions of a multiset onℕ2, finiteness of a multiset on ℕ2 and degree of a multiset on ℕ2, from Section 4 of [8]. By a lexicographic multiset onℕ2, we mean a finite multiset π={(α1,β1),…,(αt,βt)} on ℕ2 such that βk≥βk+1∀k, and if βk=βk+1, then αk≥αk+1,k=1,…,t-1. Given a lexicographic multiset π on ℕ2, let πt denote the multiset on ℕ2 (which is not necessarily lexicographic) obtained by switching the two coordinates of π. We call the multiset πt on ℕ2 the transpose of the multisetπ. Consider a pair {π1,π2} of multisets on ℕ2 (not necessarily lexicographic), where both π1 and π2 are of the same degree (say, t). Let π1={(a1,b1),…,(at,bt)}, and π2={(d1,c1),…,(dt,ct)}.We call the first coordinate of the multisetπ1 the a-cell, the second coordinate of π1 the b-cell, the first coordinate of π2 the d-cell, and the second coordinate of π2 the c-cell. Any entry in the pair {π1,π2} of multisets on ℕ2 can be identified uniquely by specifying 3 coordinates: the cell ∇ of {π1,π2} in which the entry lies (∇=a,b,c,ord), the position i (counting from left to right) of the entry in the cell ∇, and the value ⊑(∇,i) of the entry sitting in the ith position of the cell ∇.SetSπ1,π2:={x∣x=(∇,i,⊑(∇,i)),∇∈{a,b,c,d},i∈{1,…,t}}. For any x∈Sπ1,π2, let
(6)Dπ1,π2(x):={(c,t+1-i,⊑(c,t+1-i))ifx=(a,i,⊑(a,i))(d,t+1-i,⊑(d,t+1-i))ifx=(b,i,⊑(b,i))(a,t+1-i,⊑(a,t+1-i))ifx=(c,i,⊑(c,i))(b,t+1-i,⊑(b,t+1-i))ifx=(d,i,⊑(d,i)).∀i∈{1,…,t}We callDπ1,π2(x) the dual ofxwith respect to the pair{π1,π2}of multisets onℕ2. Note that, for every x∈Sπ1,π2, we have Dπ1,π2(x)∈Sπ1,π2. For any two elements x,x′∈Sπ1,π2, where x=(∇,i,⊑(∇,i)) and x′=(∇′,i′,⊑(∇′,i′)), we say that x≾x′ if ⊑(∇,i)≤⊑(∇′,i′). Similarly, we say that x⋨x′ if ⊑(∇,i)<⊑(∇′,i′) and x≈x′ if ⊑(∇,i)=⊑(∇′,i′).The above pair{π1,π2} of multisets on ℕ2 is said to be skew-symmetric lexicographic if the following conditions are satisfied simultaneously.(i)
π1 is a lexicographic multiset on ℕ2.(ii)
π2t is a lexicographic multiset on ℕ2.(iii)
ai<dt+1-i∀i∈{1,…,t}.(iv)
bi<ct+1-i∀i∈{1,…,t}.(v)
For anyx,y∈Sπ1,π2, x≾y  if and only ifDπ1,π2(x)≿Dπ1,π2(y). Moreover, x⋨y  if and only ifDπ1,π2(x)⋩Dπ1,π2(y), and x≈y`  if and only ifDπ1,π2(x)≈Dπ1,π2(y) (→ this property is called the duality property associated to the pair {π1,π2} of skew-symmetric lexicographic multisets on ℕ2).For any pair{π1,π2} of skew-symmetric lexicographic multisets on ℕ2, we define the degree of the pair to be 2 times the degree of π1 (or of π2, they are the same). A pair {π1,π2} of skew-symmetric lexicographic multisets on ℕ2 is said to be negative if ak<bk,k=1,…,t, positive if ak>bk,k=1,…,t, and nonempty if ak≠bk,k=1,…,t, where π1={(a1,b1),…,(at,bt)} and π2={(d1,c1),…,(dt,ct)}. Note that condition (v) above will imply that if ak<bk∀k=1,…,t, then dt+1-k<ct+1-k∀k=1,…,t. Similarly, if ak>bk∀k=1,…,t, then dt+1-k>ct+1-k∀k=1,…,t, and if ak≠bk∀k=1,…,t, then dt+1-k≠ct+1-k∀k=1,…,t.Let{π1,π2} be a pair of nonempty skew-symmetric lexicographic multisets on ℕ2 given by π1={(a1,b1),…,(at,bt)} and π2={(d1,c1),…,(dt,ct)}. Let us denote by π1- (resp., π1+) the lexicographic multiset on ℕ2 consisting of those elements of π1 such that ai<bi (resp., ai>bi). Let us denote by π2- (resp., π2+) the lexicographic multiset on ℕ2 consisting of those elements of π2 such that di<ci (resp., di>ci). We call {π1-,π2-} and {π1+,π2+} the negative and positive parts respectively of the pair {π1,π2}. Note here that because of condition (v) above, π1- and π2- will have the same degree, and the same holds true for π1+ and π2+. It is easy to see now that both the pairs {π1-,π2-} and {π1+,π2+} of multisets on ℕ2 are skew-symmetric lexicographic in their own right.Given a lexicographic multisetπ on ℕ2, define l(π) to be the lexicographic multiset on ℕ2 obtained by first switching the two coordinates of π and then rearranging the elements so that the new multiset is lexicographic. Let lt be a map from the set of all lexicographic multisets on ℕ2 to itself given by first switching the two coordinates of a given lexicographic multiset π and then rearranging the elements so that the transpose of the resulting multiset becomes lexicographic.We now define an involutionL on the set of all pairs of skew-symmetric lexicographic multisets on ℕ2 by L({π1,π2}):={l(π1),lt(π2)}. It is easy to check that this map is well defined, and it maps pairs of negative skew-symmetric lexicographic multisets on ℕ2 to positive ones, and vice versa. Thus L gives a bijective pairing between the set of all pairs of negative skew-symmetric lexicographic multisets on ℕ2 and the set of all pairs of positive skew-symmetric lexicographic multisets on ℕ2.


## 6. The Orthogonal-Bounded-RSK Correspondence

We next define theOrthogonal-bounded-RSK correspondence(OBRSK) as a function which maps a pair of negative skew-symmetric lexicographic multisets on ℕ2 to a negative skew-symmetric notched bitableau. Let {π1,π2} be a pair of negative skew-symmetric lexicographic multisets on ℕ2 whose entries are labeled as in Section 5. We inductively form a sequence of notched bitableaux (P(0),Q(0)), (P(1),Q(1)), …,(P(t),Q(t)), such that each (P(i),Q(i)) is of even size, and P(i) is semistandard on bi for every i=1,…,t as follows.Let(P(0),Q(0))=(∅,∅), and let b0=b1. Assume inductively that we have formed (P(i),Q(i)), such that the notched bitableau (P(i),Q(i)) is of even size, P(i) is semistandard on bi and thus on bi+1, since bi+1≤bi.Let us first fix some notation and terminology. Letpkj(i) (resp., qkj(i)) denote the entry in the kth row and jth column of P(i) (resp., Q(i)). Let 2lk(i) denote the total number of entries (note that it is always even) in the kth row of P(i) (or Q(i)).Given an arbitrary notched tableauP and any row number k of P, we call the entry in the jth box (counting from left to right) the forwardjth entry of thekth row ofP. Similarly, we call the entry in the jth box (counting from right to left) of P the backwardjth entry of thekth row ofP.It is now easy to see that the backwardjth entry of the kth row of Q(i) is actually equal to the forward (2lk(i)+1-j)th entry of Q(i). We now describe the OBRSK correspondence for the pair {π1,π2} of negative skew-symmetric lexicographic multisets on ℕ2 as mentioned above in Section 5.Perform the bounded insertion processP(i)←bi+1ai+1 as in [8]. In this finite-step process of bounded insertion, suppose that ai+1 had bumped the “forward j1th entry” of the 1st row of P(i)<bi+1 (see Section 3 of [8] for the notation P(i)<bi+1, it comes in the paragraph right after Example 3.1 of [8]); the “forward j1th entry” of the 1st row of P(i)<bi+1 has bumped the “forward j2th entry” of the 2nd row of P(i)<bi+1,… and so on until, at some point, a number is placed in a new box at the right end of some row of P(i)<bi+1, say this happens at the row number K(i) of P(i)<bi+1. Say that the entry of the new box (as mentioned in the previous statement) becomes the forward jK(i)th entry of the K(i)th row of P(i)←bi+1ai+1. We then construct a new notched tableau as follows.We letct-i bump the “backward j1th entry” of the 1st row of Q(i); then we let the “backward j1th entry” of the 1st row of Q(i) bump the “backward j2th entry” of the 2nd row of Q(i),… and so on until, at some point, a number is placed in a new box at the backward jK(i)th position of the K(i)th row of Q(i), shifting all entries in the backward 1st… up to (and including) the backward (jK(i)-1)th positions of the K(i)th row of Q(i) to the right by one box. Essentially, whatever we did for the bounded insertion process producing P(i)←bi+1ai+1, we do a dual version of the same process on Q(i) with the integer ct-i. We denote the resulting notched tableau by Q(i)←dualct-i.Note here that the tableauxP(i)←bi+1ai+1 and Q(i)←dualct-i so constructed are of the same shape, but there exists one row in both of them in which the total number of entries is odd. We wanted to construct a notched bitableau (P(i+1),Q(i+1)) inductively from (P(i),Q(i)) which should be of even size. We make it possible in the following way.LetK(i) be the row number of P(i) (or of Q(i)) at which the above-mentioned insertion algorithm had stopped. Place dt+1-(i+1) (=dt-i) in a new box at the rightmost end of the K(i)th row of P(i)←bi+1ai+1. We denote the resulting notched tableau by P(i+1). It is an easy exercise to see that P(i+1) as constructed above will be semistandard on bi+1. After this, we place bi+1 in a new box at the leftmost end of the K(i)th row of Q(i)←dualct-i, shifting all previously existing entries in the K(i)th row of Q(i)←dualct-i to the right by one box. We denote the resulting notched tableau by Q(i+1). Clearly P(i+1) and Q(i+1) have the same shape. Now we have got hold of a notched bitableau (P(i+1),Q(i+1)) which is of even size.Then OBRSK({π1,π2}) is defined to be (P(t),Q(t)). In the process above, we write (P(i+1),Q(i+1))=(P(i),Q(i))←bi+1,ct+1-(i+1)ai+1,dt+1-(i+1). In terms of this notation, OBRSK ({π1,π2})=((∅,∅)←b1,cta1,dt)⋯←bt,c1at,d1.Lemma 15.
With notation as in the definition of the OBRSK correspondence mentioned above,P(i) is row strict for all i∈{1,…,t}.Proof.
We will prove the lemma by induction oni. The base case (i.e., when i=1) of induction is easy to see. Now let i∈{1,…,t-1}. Assume inductively that P(i) is row strict. We will now prove that P(i+1) is row strict. That P(i)←bi+1ai+1 is row strict follows in the same way as in [8]. Note that P(i+1) is obtained from P(i)←bi+1ai+1 by adding dt-i at the rightmost end of some row of P(i)←bi+1ai+1, say the kth row. It now suffices to ensure that dt-i is strictly bigger than all entries in the kth row of P(i)←bi+1ai+1. It follows from the defining properties of the pair of negative skew-symmetric lexicographic multisets {π1,π2} on ℕ2 that dt-i is bigger than or equal to all entries of P(i)←bi+1ai+1. But here we need to prove something sharper; namely, dt-i is strictly bigger than all entries in the kth row of P(i)←bi+1ai+1. We will prove this now.
Clearly all the entries ofP(i)←bi+1ai+1 are contained in {a1,…,ai+1}∪˙{dt+1-i,…,dt} (see Section 4 of [8] for the notation ∪˙). Also it is easy to observe that aj<dt-i∀j∈{1,…,i+1}. So if the rightmost element of the kth row of P(i)←bi+1ai+1 equals aj for some j∈{1,…,i+1}, then we are done. Otherwise, the element in the rightmost end of the kth row of P(i)←bi+1ai+1 is dj for some j∈{t+1-i,…,t} (say j0). If dj0<dt-i, then we are done. If not, then clearly dj0=dt-i. It then follows from duality that bt+1-j0=bi+1, and it is also clear that t+1-j0<i+1.
But it is an easy exercise to check that ifl,l′∈{1,…,t} are such that l<l′ and bl=bl′, then the number of the row in which dt+1-l′ lies in P(l′) is strictly bigger than the number of the row in which dt+1-l lies in P(l′) (Here, the row number is counted from top to bottom.). So dj0 and dt-i cannot lie in the same row of P(i+1), a contradiction, hence proved.The proof of the following lemma appears in Section9.Lemma 16.
If{π1,π2} is a pair of negative skew-symmetric lexicographic multisets on ℕ2, then OBRSK({π1,π2}) is a negative skew-symmetric notched bitableau.Lemma 17.
The map OBRSK is a degree-preserving bijection from the set of all pairs of negative skew-symmetric lexicographic multisets onℕ2 to the set of all negative skew-symmetric notched bitableaux.Proof.
That OBRSK is degree preserving is obvious. To show that OBRSK is a bijection, we define its inverse, which we call thereverse of OBRSK or ROBRSK.
Note that the entire procedure used to form(P(i+1),Q(i+1)) from (P(i),Q(i)), ai+1, bi+1, ct-i and dt-i, i=1,…,t-1, is reversible. In other words, by knowing only (P(i+1),Q(i+1)), we can retrieve (P(i),Q(i)), ai+1, bi+1, ct-i, and dt-i. First, we obtain bi+1; it is the minimum entry of Q(i+1). Look at the lowest row in which bi+1 appears in Q(i+1), say it is row number s (counting from top to bottom). In the same row (row number s, counting from top to bottom) of P(i+1), look at the rightmost entry: this entry is precisely dt-i. Remove this entry (which is dt-i) from the sth row of  P(i+1), that will give us the notched tableau P(i)←bi+1ai+1. Similarly remove the leftmost entry (which is bi+1) from the sth row of Q(i+1), and all other entries in this row of Q(i+1) should be moved one box to the left: this will give us the notched tableau Q(i)←dualct-i.
Then, in thesth row of P(i)←bi+1ai+1, select the greatest entry which is less than bi+1. This entry was the new box of the bounded insertion. If we begin reverse bounded insertion with this entry, we retrieve P(i) and ai+1. Look at the path in P(i)←bi+1ai+1 starting from the sth row to the topmost row, along which this reverse bounded insertion had happened. Trace the “dual path” in Q(i)←dualct-i and do a dual of the reverse bounded insertion (which was done originally on P(i)←bi+1ai+1 to retrieve P(i) and ai+1) on Q(i)←dualct-i: that will give us Q(i) and ct-i out of Q(i)←dualct-i.
We call this process of obtaining(P(i),Q(i)), ai+1, bi+1, ct-i, and dt-i from (P(i+1),Q(i+1)) described in the paragraphs above a reverse step and denote it by (P(i),Q(i))=(P(i+1),Q(i+1))→bi+1,ct-iai+1,dt-i. We call the process of applying all the reverse steps sequentially to retrieve {π1,π2} from (P(t),Q(t)) the reverse of OBRSK or ROBRSK.
If(P(t),Q(t)) is an arbitrary negative skew-symmetric notched bitableau (which we do not assume to be OBRSK ({π1,π2}) for some {π1,π2}), then we can still apply a sequence of reverse steps to (P(t),Q(t)), to sequentially obtain (P(i),Q(i)), ai+1, bi+1, ct-i, and dt-i, i=t-1,…,1. For this process to be well defined, however, it must first be checked that the successive (P(i),Q(i)) are negative skew-symmetric notched bitableaux. For this, it suffices to prove a statement very similar to that proved in Lemma 16; namely, “If (P,Q) is a negative skew-symmetric notched bitableau, then (P′,Q′):=(P,Q)→b,ca,d is a negative skew-symmetric notched bitableau, a<b, d<c, a<d,b<c are positive integers, d is greater than or equal to all entries of P, and b is less than or equal to all entries of Q.” That a<b, d<c, a<d, and b<c are positive integers, d is greater than or equal to all entries of P, and b is less than or equal to all entries of Q follows immediately from the definition of a reverse step. That (P′,Q′) is a negative skew-symmetric notched bitableau follows in much the same manner as the proof of Lemma 16; we omit the details.
It remains to be shown that the pair of multisets onℕ2 produced by applying this sequence of reverse steps to the arbitrary skew-symmetric notched bitableau (P(t),Q(t)) is skew-symmetric lexicographic. The proof of this uses the duality property of skew-symmetric notched bitableaux, the facts mentioned in the preceding paragraph regarding the integers a,b,c, and d, and the rest of the proof goes similarly as in the proof of Lemma 6.3 of [8].
At each step, OBRSK and the reverse of ROBRSK are inverse to each other. Thus they are inverse maps.The map OBRSK can be extended to all pairs of nonempty skew-symmetric lexicographic multisets onℕ2. If {π1,π2} is a pair of positive skew-symmetric lexicographic multisets on ℕ2, then define OBRSK({π1,π2}) to be ι(OBRSK(L({π1,π2}))), where the definition of the map ι is given at the end of Section 4. If {π1,π2} is a pair of nonempty skew-symmetric lexicographic multisets on ℕ2, with negative and positive parts {π1-,π2-} and {π1+,π2+}, then define OBRSK({π1,π2}) to be the skew-symmetric notched bitableau whose negative and positive parts are OBRSK ({π1-,π2-}) and OBRSK ({π1+,π2+}).Example 18.
Letπ1={(4,17),(3,17),(3,14),(7,10),(4,9)}, and π2={(20,25),(19,22),(15,26),(12,26),(12,25)}. See Figure 2.
Therefore.(7)


## 7. Restricting the OBRSK Correspondence

### 7.1. Restricting byβ

Letβ∈I(d). We say that a skew-symmetric notched bitableau(P,Q)   is onβ¯×β, if all entries of P are in β¯, all entries of Q are in β, and the sum of any entry in P (or in Q) with its dual (with respect to (P,Q)) is 2d+1.Given any monomialU in 𝔒ℜ(β), we can define a monomial U# in 𝔄ℜ(β) as follows: U#:={(c*,r*)|(r,c)∈U}. We say that a pair {V1,V2} of skew-symmetric lexicographic multisets on ℕ2 is a pair of skew-symmetric lexicographic multisets onβ¯×β, if V1 is a monomial in 𝔒ℜ(β), V2 is a monomial in 𝔄ℜ(β), a number of elements (counting multiplicities) in V1 and V2 are the same, and V2=V1#. It is easy to see from the construction of OBRSK that, if {U,U#} is a pair of nonempty skew-symmetric lexicographic multisets on β¯×β, then OBRSK ({U,U#}) is a nonvanishing skew-symmetric notched bitableau on β¯×β, and vice versa. Thus we obtain the following.Corollary 19.
The map OBRSK restricts to a degree-preserving bijection from the set of all pairs of nonempty (resp., negative, positive) skew-symmetric lexicographic multisets onβ¯×β to the set of all nonvanishing (resp., negative, positive) skew-symmetric notched bitableaux on β¯×β.


### 7.2. Restricting byT and W

LetT and W be negative and positive subsets of ℕ2, respectively, satisfying the condition that:
(8)T(1),T(2),W(1),andW(2)aresubsetsofℕ,
where the notation T(1),T(2),W(1), and W(2) is explained in Section 4 of [8]. In this subsection, we will define the boundedness of a nonempty pair of skew-symmetric lexicographic multisets on ℕ2 by T,W, and this is given in Definition 26, after some definitions which are necessary to understand it.Definition 20.
Adual pair of chains in ℕ2 is a pair of subsets {C1={(e1,f1),…,(em,fm)},C2={(g1,h1),…,(gm,hm)}} such that C1 is a chain in ℕ2 in the sense of Section 7 of [8], and {σ1,σ2} is a pair of skew-symmetric lexicographic multisets on ℕ2, where σ1={(e1,f1),…,(em,fm)} and σ2={(g1,h1),…,(gm,hm)}.Definition 21.
Let{U1,U2} be a pair of skew-symmetric lexicographic multisets on ℕ2. Let {C1,C2} be a dual pair of chains in ℕ2 such that Ci is contained in the underlying set of Ui for all i=1,2. Given any element in C1 (say, the ith element counting from left to right), look at the first (counting from left to right) element in U1 which is entrywise the same as the ith element of C1. Call this element of U1 the iminth element. Let t be the total number of elements in U1. One calls the (t+1-imin)th element of U2 (counting from left to right) the dual element in U2 corresponding to the ith element of C1.Definition 22.
Given any pair{π1,π2} of skew-symmetric lexicographic multisets on ℕ2, where π1={(a1,b1),…,(at,bt)} and π2={(d1,c1),…,(dt,ct)}, one says that the elements (ai,bi) and (dt+1-i,ct+1-i) of π1 and π2, respectively, are dual to each other with respect to {π1,π2}.Definition 23.
Given a pair of{U1,U2} of skew-symmetric lexicographic multisets on ℕ2, one says that a dual pair {C1,C2} of chains in ℕ2 is a dual pair of chains in{U1,U2}, if the following two conditions are satisfied simultaneously: (i) Ci is contained in the underlying set of Ui for all i=1,2. (ii) Given any element of C1 (say, the ith element counting from left to right), the dual element in U2 corresponding to it is entrywise the same as the dual element of the ith element of C1 with respect to {C1,C2}.Definition 24.
Given any row-strict notched bitableau(P,Q), we associate with it 2 subsets of ℕ2 as follows. Let P1 and Q1 denote the topmost rows of P and Q, respectively. Let p11<⋯<p1k1 and q11<⋯<q1k1 denote the entries of P1 and Q1, respectively. We denote by (P,Q)up the subset of ℕ2 given by {(p11,q11),…,(p1k1,q1k1)}. We denote by (P,Q)down the subset of ℕ2 obtained similarly if we work with the lower-most rows of P and Q, instead of the topmost rows. See Example 25 for an illustration.Example 25.
Consider the row-strict notched bitableau(9)
and call it (𝒫,𝒬). Then (𝒫,𝒬)up={(H,D),(J,A),(T,K),(Q,M)}, and (𝒫,𝒬)down={(F,C),(R,O)}.Definition 26.
A nonempty pair of skew-symmetric lexicographic multisets{U1,U2} on ℕ2 is said to be bounded byT,W if for every dual pair {C1,C2} of chains in {U1,U2} one has
(10)T≤(PC1-,C2-,QC1-,C2-)up,(PC1+,C2+,QC1+,C2+)down≤W,
where one uses the order on multisets on ℕ2 defined in Section 4 of [8]; (PC1-,C2-,QC1-,C2-) is defined to be OBRSK ({C1-,C2-}), and (PC1+,C2+,QC1+,C2+) is defined as OBRSK ({C1+,C2+}).It is worthwhile to note that(PC1-,C2-,QC1-,C2-)up, (PC1+,C2+,QC1+,C2+)down, T, and W are subsets of ℕ2, and they are not pairs of subsets!With this definition, the OBRSK correspondence is a bounded function, in the sense that it maps bounded sets to bounded sets. More precisely, we have the following Lemma (to understand the statement of this lemma, we need to recall the notion of a semistandard notched bitableau being bounded byT,W from Section 5 of [8]), whose proof appears in Section 9.Lemma 27.
If a pair{U1,U2} of nonempty skew-symmetric lexicographic multisets on ℕ2 is bounded by T,W, then OBRSK({U1,U2}) is bounded by T,W.Definition 28.
A dual pair{C1,C2} of chains in ℕ2 is called a dual pair of chains inβ¯×β if {C1,C2} is a pair of skew-symmetric lexicographic multisets on β¯×β.Remark 29.
A general dual pair of chains inβ¯×β will look like {C,C#} for some extended β-chain C in 𝔒ℜ(β). Note also that if {U1,U2} is a pair of skew-symmetric lexicographic multisets on β¯×β, that is, if U2=U1#, then any dual pair of chains {C1,C2} in {U1,U2} must be a dual pair of chains in β¯×β; in other words, we must have C2=C1#, where C1 is an extended β-chain in 𝔒ℜ(β).Remark 30.
LetT and W be negative and positive subsets of ℕ2, respectively, satisfying (8). A nonempty pair of skew-symmetric lexicographic multisets {U,U#} on β¯×β is said to be bounded byT,W if, for every dual pair of chains {C,C#} in β¯×β which is contained in the underlying set of {U,U#},
(11)T≤(PC-,C-#,QC-,C-#)up,(PC+,C+#,QC+,C+#)down≤W,
where we use the order on multisets on ℕ2 defined in Section 4 of [8], and (PC-,C-#,QC-,C-#) (resp., (PC+,C+#,QC+,C+#)) is defined to be OBRSK ({C-,C-#}) (resp., OBRSK ({C+,C+#})). It is worthwhile to note that (PC-,C-#,QC-,C-#)up, (PC+,C+#,QC+,C+#)down, T, and W are subsets of ℕ2; they are not pairs of subsets!LetT and W be negative and positive subsets of β¯×β, respectively, satisfying (8). Combining Corollary 19 and Lemma 27, we obtain the following.Corollary 31.
For any positive integerm, the number of pairs of nonempty skew-symmetric lexicographic multisets on β¯×β bounded by T,W of degree 2m is less than or equal to the number of nonvanishing skew-symmetric notched bitableaux on β¯×β bounded by T,W of degree 2m.


## 7.1. Restricting byβ

Letβ∈I(d). We say that a skew-symmetric notched bitableau(P,Q)   is onβ¯×β, if all entries of P are in β¯, all entries of Q are in β, and the sum of any entry in P (or in Q) with its dual (with respect to (P,Q)) is 2d+1.Given any monomialU in 𝔒ℜ(β), we can define a monomial U# in 𝔄ℜ(β) as follows: U#:={(c*,r*)|(r,c)∈U}. We say that a pair {V1,V2} of skew-symmetric lexicographic multisets on ℕ2 is a pair of skew-symmetric lexicographic multisets onβ¯×β, if V1 is a monomial in 𝔒ℜ(β), V2 is a monomial in 𝔄ℜ(β), a number of elements (counting multiplicities) in V1 and V2 are the same, and V2=V1#. It is easy to see from the construction of OBRSK that, if {U,U#} is a pair of nonempty skew-symmetric lexicographic multisets on β¯×β, then OBRSK ({U,U#}) is a nonvanishing skew-symmetric notched bitableau on β¯×β, and vice versa. Thus we obtain the following.Corollary 19.
The map OBRSK restricts to a degree-preserving bijection from the set of all pairs of nonempty (resp., negative, positive) skew-symmetric lexicographic multisets onβ¯×β to the set of all nonvanishing (resp., negative, positive) skew-symmetric notched bitableaux on β¯×β.


## 7.2. Restricting byT and W

LetT and W be negative and positive subsets of ℕ2, respectively, satisfying the condition that:
(8)T(1),T(2),W(1),andW(2)aresubsetsofℕ,
where the notation T(1),T(2),W(1), and W(2) is explained in Section 4 of [8]. In this subsection, we will define the boundedness of a nonempty pair of skew-symmetric lexicographic multisets on ℕ2 by T,W, and this is given in Definition 26, after some definitions which are necessary to understand it.Definition 20.
Adual pair of chains in ℕ2 is a pair of subsets {C1={(e1,f1),…,(em,fm)},C2={(g1,h1),…,(gm,hm)}} such that C1 is a chain in ℕ2 in the sense of Section 7 of [8], and {σ1,σ2} is a pair of skew-symmetric lexicographic multisets on ℕ2, where σ1={(e1,f1),…,(em,fm)} and σ2={(g1,h1),…,(gm,hm)}.Definition 21.
Let{U1,U2} be a pair of skew-symmetric lexicographic multisets on ℕ2. Let {C1,C2} be a dual pair of chains in ℕ2 such that Ci is contained in the underlying set of Ui for all i=1,2. Given any element in C1 (say, the ith element counting from left to right), look at the first (counting from left to right) element in U1 which is entrywise the same as the ith element of C1. Call this element of U1 the iminth element. Let t be the total number of elements in U1. One calls the (t+1-imin)th element of U2 (counting from left to right) the dual element in U2 corresponding to the ith element of C1.Definition 22.
Given any pair{π1,π2} of skew-symmetric lexicographic multisets on ℕ2, where π1={(a1,b1),…,(at,bt)} and π2={(d1,c1),…,(dt,ct)}, one says that the elements (ai,bi) and (dt+1-i,ct+1-i) of π1 and π2, respectively, are dual to each other with respect to {π1,π2}.Definition 23.
Given a pair of{U1,U2} of skew-symmetric lexicographic multisets on ℕ2, one says that a dual pair {C1,C2} of chains in ℕ2 is a dual pair of chains in{U1,U2}, if the following two conditions are satisfied simultaneously: (i) Ci is contained in the underlying set of Ui for all i=1,2. (ii) Given any element of C1 (say, the ith element counting from left to right), the dual element in U2 corresponding to it is entrywise the same as the dual element of the ith element of C1 with respect to {C1,C2}.Definition 24.
Given any row-strict notched bitableau(P,Q), we associate with it 2 subsets of ℕ2 as follows. Let P1 and Q1 denote the topmost rows of P and Q, respectively. Let p11<⋯<p1k1 and q11<⋯<q1k1 denote the entries of P1 and Q1, respectively. We denote by (P,Q)up the subset of ℕ2 given by {(p11,q11),…,(p1k1,q1k1)}. We denote by (P,Q)down the subset of ℕ2 obtained similarly if we work with the lower-most rows of P and Q, instead of the topmost rows. See Example 25 for an illustration.Example 25.
Consider the row-strict notched bitableau(9)
and call it (𝒫,𝒬). Then (𝒫,𝒬)up={(H,D),(J,A),(T,K),(Q,M)}, and (𝒫,𝒬)down={(F,C),(R,O)}.Definition 26.
A nonempty pair of skew-symmetric lexicographic multisets{U1,U2} on ℕ2 is said to be bounded byT,W if for every dual pair {C1,C2} of chains in {U1,U2} one has
(10)T≤(PC1-,C2-,QC1-,C2-)up,(PC1+,C2+,QC1+,C2+)down≤W,
where one uses the order on multisets on ℕ2 defined in Section 4 of [8]; (PC1-,C2-,QC1-,C2-) is defined to be OBRSK ({C1-,C2-}), and (PC1+,C2+,QC1+,C2+) is defined as OBRSK ({C1+,C2+}).It is worthwhile to note that(PC1-,C2-,QC1-,C2-)up, (PC1+,C2+,QC1+,C2+)down, T, and W are subsets of ℕ2, and they are not pairs of subsets!With this definition, the OBRSK correspondence is a bounded function, in the sense that it maps bounded sets to bounded sets. More precisely, we have the following Lemma (to understand the statement of this lemma, we need to recall the notion of a semistandard notched bitableau being bounded byT,W from Section 5 of [8]), whose proof appears in Section 9.Lemma 27.
If a pair{U1,U2} of nonempty skew-symmetric lexicographic multisets on ℕ2 is bounded by T,W, then OBRSK({U1,U2}) is bounded by T,W.Definition 28.
A dual pair{C1,C2} of chains in ℕ2 is called a dual pair of chains inβ¯×β if {C1,C2} is a pair of skew-symmetric lexicographic multisets on β¯×β.Remark 29.
A general dual pair of chains inβ¯×β will look like {C,C#} for some extended β-chain C in 𝔒ℜ(β). Note also that if {U1,U2} is a pair of skew-symmetric lexicographic multisets on β¯×β, that is, if U2=U1#, then any dual pair of chains {C1,C2} in {U1,U2} must be a dual pair of chains in β¯×β; in other words, we must have C2=C1#, where C1 is an extended β-chain in 𝔒ℜ(β).Remark 30.
LetT and W be negative and positive subsets of ℕ2, respectively, satisfying (8). A nonempty pair of skew-symmetric lexicographic multisets {U,U#} on β¯×β is said to be bounded byT,W if, for every dual pair of chains {C,C#} in β¯×β which is contained in the underlying set of {U,U#},
(11)T≤(PC-,C-#,QC-,C-#)up,(PC+,C+#,QC+,C+#)down≤W,
where we use the order on multisets on ℕ2 defined in Section 4 of [8], and (PC-,C-#,QC-,C-#) (resp., (PC+,C+#,QC+,C+#)) is defined to be OBRSK ({C-,C-#}) (resp., OBRSK ({C+,C+#})). It is worthwhile to note that (PC-,C-#,QC-,C-#)up, (PC+,C+#,QC+,C+#)down, T, and W are subsets of ℕ2; they are not pairs of subsets!LetT and W be negative and positive subsets of β¯×β, respectively, satisfying (8). Combining Corollary 19 and Lemma 27, we obtain the following.Corollary 31.
For any positive integerm, the number of pairs of nonempty skew-symmetric lexicographic multisets on β¯×β bounded by T,W of degree 2m is less than or equal to the number of nonvanishing skew-symmetric notched bitableaux on β¯×β bounded by T,W of degree 2m.


## 8. The Initial Ideal

In this section, we will prove the main result of the paper, that is, Theorem7.Proof of Theorem7.
We wish to show thatin▹I=〈Chainsαγ(β)〉. Since we already know from Remark 12 that Chainsαγ(β)⊆in▹I, it follows that 〈Chainsαγ(β)〉⊆in▹I. Hence it suffices to show that in▹I⊆〈Chainsαγ(β)〉 or equivalently that, for any m≥1, cardinality of the set of all degree m monomials in P/〈Chainsαγ(β)〉 is less than or equal to cardinality of the set of all degree m monomials in P/in▹I. Now since standard monomials on Yα,βγ (standard monomials on Yα,βγ is defined in Definition 32) and the monomials in P/in▹I both induce homogeneous bases for P/I, it suffices to show that, for any m≥1, cardinality of the set of all degree m monomials in P/〈Chainsαγ(β)〉 is less than or equal to cardinality of the set of all degree m standard monomials on Yα,βγ.
We wish to give a different indexing set for the standard monomials onYα,βγ. Let Iβ(skew-symm) denote the set of all pairs (R,S) such that all of the following conditions are satisfied.(i)
R⊂β¯.(ii)
S⊂β.(iii)
|R|=|S|, and this cardinality is even.(iv)
IfR={r1<⋯<r2l} and S={s1<⋯<s2l}, then ri+s2l+1-i=2d+1∀i∈{1,…,2l}.
DefiningR-S:=R∪˙(β∖S) (see Section 4 of [8]), we have the following fact, which is easily verified.
The map(R,S)↦R-S is a bijection fromIβ (skew-symm)  toI(d), (Indeed, the inverse map is given by θ↦(θ∖β,β∖θ).).
Note that under this bijection,(∅,∅) maps to β. Let (Rα,Sα) and (Rγ,Sγ) be the preimages of the elements α and γ (of I(d)), respectively. Define Tα and Wγ to be any subsets of β¯×β such that (Tα)(1)=Rα, (Tα)(2)=Sα, (Wγ)(1)=Rγ, and (Wγ)(2)=Sγ. Observe that Tα and Wγ satisfy (8).
Under this identification ofIβ (skew-symm) with I(d), the inequalities which define nonvanishing skew-symmetric notched bitableaux on β¯×β bounded by Tα, Wγ are precisely the inequalities which define the standard monomials on Yα,βγ (the inequalities (12), (13), and (14) of this paper). Thus we obtain that the degree 2m nonvanishing skew-symmetric notched bitableaux on β¯×β bounded by Tα,Wγ forms an indexing set for the degree m standard monomials on Yα,βγ. Hence it suffices to show that, for any m≥1, cardinality of the set of all degree m monomials in P/〈Chainsαγ(β)〉 is less than or equal to cardinality of the set of all nonvanishing skew-symmetric notched bitableaux on β¯×β bounded by Tα,Wγ of degree 2m.
On the other hand, it is easy to see that the setChainsαγ(β) can be rewritten in the following format: Chainsαγ(β) is the set of all XC, where C is any nonempty extended β-chain in 𝔒ℜ(β) such that either (i)′ or (ii)′ below holds.(i)′
The chainC- is nonempty, and Rα-Sα≰(PC-,C-#,QC-,C-#)(1)up-(PC-,C-#,QC-,C-#)(2)up.(ii)′
The chainC+ is nonempty, and (PC+,C+#,QC+,C+#)(1)down-(PC+,C+#,QC+,C+#)(2)down≰Rγ-Sγ.
It is now obvious that the pairs of nonempty skew-symmetric lexicographic multisets onβ¯×β bounded by Tα,Wγ of degree 2m form an indexing set for the degree m monomials of P/〈Chainsαγ(β)〉. Hence we have that, for any m≥1, cardinality of the set of all degree m monomials in P/〈Chainsαγ(β)〉 is equal to cardinality of the set of all pairs of nonempty skew-symmetric lexicographic multisets on β¯×β bounded by Tα,Wγ of degree 2m.
Again from Corollary31, it follows that for any m≥1, cardinality of the set of all pairs of nonempty skew-symmetric lexicographic multisets on β¯×β bounded by Tα,Wγ of degree 2m is less than or equal to cardinality of the set of all nonvanishing skew-symmetric notched bitableaux on β¯×β bounded by Tα,Wγ of degree 2m. Thus 〈Chainsαγ(β)〉⊇in▹I.Definition 32.
LetP=𝔨[X(r,c)|(r,c)∈𝔒ℜ(β)]. Recall the concept of a Pfaffian (denoted by fθ,β for θ∈I(d)) from Section 3.3 of this paper. One calls f=fθ1,β⋯fθr,β∈P a standard monomial if θ1,…,θr∈I(d),
(12)θ1≤⋯≤θr
and for each i∈{1,…,r}, either
(13)θi<βorθi>β.
If in addition, for α,γ∈I(d),
(14)α≤θ1,θr≤γ,
then we say that f is standard onYα,βγ. We define the degree of the standard monomial fθ1,β⋯fθr,β to be the sum of the β-degrees of θ1,…,θr, where for any θ∈I(d); the β-degree is defined to be one-half the cardinality of θ∖β.Remark 33.
In general, a standard monomial is not a monomial in the affine coordinatesX(r,c),(r,c)∈𝔒ℜ(β); rather, it is a polynomial. It is only a monomial in the fθ,β’s.


## 9. Proofs

In this section, we give proofs of Lemmas16 and 27. To give a proof of Lemma 16, we need Lemma 34 mentioned below whose proof is left to the reader as an exercise.Lemma 34.
IfA and B are two finite multisets on ℕ of the same degree such that A≤B (where the termwise order ≤ on multisets on ℕ and the degree of a multiset on ℕ are as defined in Section 4 of [8]) and p,q are natural numbers such that p≤q, then A∪˙{p}≤B∪˙{q}.


### 9.1. Proof of Lemma16

Proof.
The proof is by induction. The base case of induction is easy to see, and hence we omit the details. Let{π1(t-1),π2(t-1)} be a pair of negative skew-symmetric lexicographic multisets on ℕ2, where π1(t-1)={(a1,b1),…,(at-1,bt-1)} and π2(t-1)={(d2,c2),…,(dt,ct)}. Let (P,Q)=OBRSK({π1(t-1),π2(t-1)}). Assume inductively that (P,Q) is a negative skew-symmetric notched bitableau. Now let {π1(t),π2(t)} be a pair of negative skew-symmetric lexicographic multisets on ℕ2 such that π1(t)=π1 and π2(t)=π2, where π1 and π2 are given by π1={(a1,b1),…,(at,bt)} and π2={(d1,c1),…,(dt,ct)}. Let (P′,Q′)=OBRSK({π1(t),π2(t)}). It suffices to show that (P′,Q′) is also a negative skew-symmetric notched bitableau.
We will first prove that(P′,Q′) is a negative semistandard notched bitableau. The fact that P′ is row strict follows from Lemma 15. It then follows from duality that Q′ is also row strict. Hence (P′,Q′) is row strict. Let r′ be the total number of rows of P′(orQ′). Let Pi′,Qi′,Pi, and Qi denote the multiset consisting of all elements in the ith row of P′,Q′,P, and Q, respectively. In fact, these multisets (Pi′,Qi′,Pi,Qi) happen to be sets because P′,Q′,P, and Q are row strict. It needs to be shown that Pi′-Qi′≤Pi+1′-Qi+1′ for 1≤i≤r′-1, and Pr′′-Qr′′⋖∅ (see Section 4 of [8] for the notation ⋖). We will prove the former statement first. To prove that Pi′-Qi′≤Pi+1′-Qi+1′ for 1≤i≤r′-1, there will be three nontrivial possibilities which will be registered below as Cases 1, 2, and 3, respectively.
Case1. at and d1 are added to the first row of P, and c1 and bt are added to the first row of Q. All rows other than the 1st row of P and Q remain unchanged.
In this case, it suffices to show thatP1∪˙Q2∪˙{at,d1}≤P2∪˙Q1∪˙{c1,bt}. It follows from induction hypothesis that P1∪˙Q2≤P2∪˙Q1. Since {π1(t),π2(t)} is a pair of negative skew-symmetric lexicographic multisets on ℕ2, we have at<bt and d1<c1. Therefore at≤bt and d1≤c1. These facts together with Lemma 34 (applied twice) imply easily that P1∪˙Q2∪˙{at,d1}≤P2∪˙Q1∪˙{c1,bt}. Hence we are done in this case.
Case2.xp bumps yp from Pi, yp bumps zp from Pi+1, and the dual bumping happens on Qi and Qi+1. Let us express the dual bumping by saying that xq bumps yq from Qi and yq bumps zq from Qi+1.
Clearly then,xp≤yp≤zp<bt, and hence xq≥yq≥zq by the property (ii) in the definition of a skew-symmetric notched bitableau. Again since all entries in Q are ≥bt, we get that xq≥yq≥zq≥bt.
We need to show thatPi′-Qi′≤Pi+1′-Qi+1′, or, in other words, Pi′∪˙Qi+1′≤Pi+1′∪˙Qi′. Note that Pi′∪˙Qi+1′=[(Pi∪˙Qi+1)∖{yp,zq}]∪˙{xp,yq} and Pi+1′∪˙Qi′=[(Pi+1∪˙Qi)∖{zp,yq}]∪˙{xq,yp}. It suffices to show that (Pi∪˙Qi+1)∖{yp,zq}≤(Pi+1∪˙Qi)∖{zp,yq} because if we can prove this much, then since xp≤yp and xq≥yq we can apply Lemma 34 and get done with the proof as in Case 1.
We will now prove that(Pi∪˙Qi+1)∖{yp,zq}≤(Pi+1∪˙Qi)∖{zp,yq}. Since zp<bt and all entries of Qi are ≥bt, therefore zp is the smallest element in Pi+1∪˙Qi which is ≥yp. It then follows from duality that zq is the biggest element in Pi∪˙Qi+1, that is, ≤yq. Also since yp<bt≤zq, we have yp<zq, and hence by duality yq>zp. Also by induction hypothesis, Pi∪˙Qi+1≤Pi+1∪˙Qi. All these facts put together prove the required thing easily, and we are done in this case.
Case3. xp bumps yp from Pi, and yp along with d1 are added to Pi+1. The dual phenomenon happens with Qi and Qi+1; we express the dual phenomenon by saying that xq bumps yq from Qi, and yq along with bt are added to Qi+1.
Clearly then,xp≤yp<bt, and hence xq≥yq by the property (ii) in the definition of a Skew-symmetric notched bitableau. Again since all entries in Q are ≥bt, we get that xq≥yq≥bt. Also yp<d1 since the position of d1 is at the rightmost end of that row of Pi+1 in which yp is placed. Hence by duality, we have yq>bt.
We need to show thatPi′-Qi′≤Pi+1′-Qi+1′, or, in other words, Pi′∪˙Qi+1′≤Pi+1′∪˙Qi′. Note that Pi′∪˙Qi+1′=(Pi∪˙Qi+1∖{yp})∪˙{bt}∪˙{xp,yq}, and Pi+1′∪˙Qi′=(Pi+1∪˙Qi∖{yq})∪˙{d1}∪˙{xq,yp}. Using Lemma 34, it is enough to prove that (Pi∪˙Qi+1∖{yp})∪˙{bt}≤(Pi+1∪˙Qi∖{yq})∪˙{d1}, which we will do now.
LetPi∪˙Qi+1 be given by X1≤⋯≤X2n, and let Pi+1∪˙Qi be given by Y1≤⋯≤Y2n (since the bitableau (P,Q) is of even size, the cardinality of Pi∪˙Qi+1 or of Pi+1∪˙Qi is even). Note that there are no elements of Pi+1 which are ≥yp and <bt. Also since bt is ≤ all elements of Qi, therefore we can conclude that there are no elements of Pi+1∪˙Qi which are ≥yp and <bt. It follows from duality that there are no elements of Pi∪˙Qi+1 which are ≤yq as well as >d1. Let (Pi∪˙Qi+1∖{yp})∪˙{bt} be given by
(15)X1≤⋯≤Xi-1≤Xi+1≤⋯≤Xj≤bt≤Xj+1≤⋯≤X2n
and let (Pi+1∪˙Qi∖{yq})∪˙{d1} be given by
(16)Y1≤⋯≤Y2n-j≤d1≤Y2n-j+1≤⋯≤Y2n-i≤Y2n-i+2≤⋯≤Y2n.
This means that yp=Xi and its dual yq=Y2n-i+1 (the ith element counting from right to left in the ordered sequence Y1≤⋯≤Y2n). Also since yp=Xi and yp<bt, it follows that i<j. Observe that bt is the jth element (counting from left to right) in the ordered sequence X1≤⋯≤Xi-1≤Xi+1≤⋯≤Xj≤bt≤Xj+1≤⋯≤X2n of integers and d1 (the dual of bt with respect to the pair {π1,π2} of multisets) is the jth element (counting from right to left) in the ordered sequence Y1≤⋯≤Y2n-j≤d1≤Y2n-j+1≤⋯≤Y2n-i≤Y2n-i+2≤⋯≤Y2n of integers. Consider any element in (Pi∪˙Qi+1∖{yp})∪˙{bt}. It can either be Xk for some k≠i or bt. If it is Xk for some k≠i, then we either need to show that Xk≤Yk, Xk≤d1, or Xk≤Yk-1 (these are the only 3 possibilities; the last possibility Xk≤Yk-1 happens if and only if k∈{2n-j+2,…,2n-i+1}). If it is bt, then we need to show that bt≤ the jth element (counting from left to right) of (Pi+1∪˙Qi∖{yq})∪˙{d1}, which can either be Yj, d1, or Yj-1.
After all these observation(s), the proof follows easily using the inequalitiesi<j, xp≤yp<bt, xq≥yq≥bt, yp<d1, and yq>bt and the facts that there are no elements of Pi+1∪˙Qi which lie in the interval [yp,bt) and that there are no elements of Pi∪˙Qi+1 which lie in the interval (d1,yq].
We have now proved that(P′,Q′) is a semistandard notched bitableau. To prove that the semistandard notched bitableau (P′,Q′) is negative, it is enough to prove that Pr′′-Qr′′⋖∅, where r′ denotes the total number of rows of P′(orQ′). If Pr′′=Pr′ and Qr′′=Qr′, then since (P,Q) is assumed to be a negative skew-symmetric notched bitableau, it follows immediately that Pr′′-Qr′′⋖∅. Otherwise, we do the following: let Pr′′ and Qr′′ be given by λ1<⋯<λs′ and δ1<⋯<δs′, respectively. It follows easily from the way the OBRSK algorithm works that there exists at least one entry in Pr′′ which is <bt. Let λl denote the largest entry in Pr′′ which is <bt. Since bt≤δ1, it now follows easily that λj<δj∀j∈{1,…,l}. Since λl is the largest entry in Pr′′ which is <bt, therefore l≥1, and it follows from duality that δs′+1-l=δs′-(l-1)>d1. Hence δs′≥δs′-(l-1)>d1=λs′. It only remains to show that if l+1<s′, then λj<δj∀j∈{l+1,…,s′-1}. Clearly since l≥1 and l+1<s′, it follows that, in this case, the last row of P (resp., Q) is the r′th row, namely Pr′ (resp., Qr′), and δs′-(l-1) is the new box of the dual insertion in Qr′. Since λs′=d1,δs′-(l-1)>d1, Pr′-Qr′⋖∅ by induction hypothesis and λj<λs′∀j∈{l+1,…,s′-1}, it is now easy to see that λj<δj∀j∈{l+1,…,s′-1}.
Hence we have proved that(P′,Q′) is a negative semistandard notched bitableau. The fact that (P′,Q′) is skew-symmetric is easy to see from the very construction of the OBRSK algorithm and from the fact that the pair {π1(t),π2(t)} of multisets is skew-symmetric, lexicographic.


### 9.2. Proof of Lemma27

Proof (proof of Lemma27).
LetT be a negative subset of ℕ2 such that T(1),T(2) are subsets of ℕ. Suppose that {U1={(a1,b1),…,(at,bt)},U2={(d1,c1),…,(dt,ct)}} is a pair of negative skew-symmetric lexicographic multisets on ℕ2. For each m=1,…,t, let U1(m):={(a1,b1),…,(am,bm)} and U2(m):={(dt+1-m,ct+1-m),…,(dt,ct)}. Let (P(m),Q(m)):=OBRSK({U1(m),U2(m)}). For proving Lemma 27, it suffices to show that if {U1,U2} is bounded by T,∅, then (P(m),Q(m)) is bounded by T,∅ for all m=1,…,t because then the proof of Lemma 27 will follow similarly as the proofs of parts (iii), (iv), and (v) of  Lemma  11.1 in [8].
Assume that{U1,U2} is bounded by T,∅. For every m∈{1,…,t}, let P1(m) (resp., Q1(m)) denote the topmost row of P(m) (resp., Q(m)). We need to show that T(1)-T(2)≤P1(m)-Q1(m). Equivalently, we need to show that for all positive integers z, |(T(1)-T(2))≤z|≥|(P1(m)-Q1(m))≤z|, where we use the definition A-B:=A∪˙(ℕ∖B) for any two subsets A and B of ℕ (see Section 4 of [8]). We will prove this by induction on m. Let us first see the base case of induction. When m=1, U1(m)={(a1,b1)} and U2(m)={(dt,ct)}. Let C1={(a1,b1)} and C2={(dt,ct)}. Since {U1,U2} is bounded by T,∅, it follows that for the dual pair {C1,C2} of chains in {U1,U2}, we have T≤(PC1-,C2-,QC1-,C2-)up≤∅. But it can be easily seen that (PC1-,C2-,QC1-,C2-)up={(a1,b1),(dt,ct)}. This proves the base case of induction. Fix an integer k∈{1,…,t-1}, and assume that (P(m),Q(m)) is bounded by T,∅ for all m≤k. We will now show that (P(k+1),Q(k+1)) is bounded by T,∅.
Before we begin the proof, let us fix some notation for the rest of this subsection. Define{p1(k),…,p2ck(k)} to be the topmost row of P(k) and {q1(k),…,q2ck(k)} the topmost row of Q(k), both listed in increasing order. Let (P,Q)=(P(k),Q(k)), a=ak+1, b=bk+1, c=ct-k, d=dt-k, (P′,Q′)=(P(k+1),Q(k+1)), {p1,…,p2c^}={p1(k),…,p2ck(k)}, and {q1,…,q2c^}={q1(k),…,q2ck(k)}. Note that {p1,…,p2c^}⊂{a1,…,ak}∪˙{dt+1-k,…,dt} and {q1,…,q2c^}⊂{b1,…,bk}∪˙{ct+1-k,…,ct}. Let P1 (resp., Q1) denote the topmost row of P (resp., Q). Similarly let P1′ (resp., Q1′) denote the topmost row of P′ (resp., Q′). We consider two cases corresponding to the two ways in which (P1′,Q1′) can be obtained from (P1,Q1).
Case1. P1′ is obtained by a bumping pl in P1, for some 1≤l≤2c^, that is, P1′=(P1∖{pl})∪˙{a} and Q1′=(Q1∖{q2c^+1-l})∪˙{c}.
Sinceb is less than or equal to all elements of {b1,…,bk} and bi<ct+1-i∀i∈{1,…,t}, it follows that b≤ all elements of {b1,…,bk}∪˙{ct+1-k,…,ct}. Therefore a<b≤q1, and hence by duality c>d≥p2c^. The fact that a bumps pl implies that a≤pl and pl<b. Hence a≤pl<b≤q1, and therefore by duality, we have c≥q2c^+1-l>d≥p2c^. For z<a, pl≤z<q2c^+1-l, or z≥c,
(17)|(T(1)-T(2))≤z|≥|(P1-Q1)≤z|=|(P1′-Q1′)≤z|,
where the first inequality follows from induction hypothesis, and second equality follows from the facts that pl-1<a≤pl<b≤q1≤q2c^+1-l≤c and p2c^≤d<c. If a=pl, then we are done. Thus we assume that a<pl (and hence by duality that c>q2c^+1-l). We now need to consider only two possible positions of z, namely, a≤z<pl and q2c^+1-l≤z<c. We claim that, for z such that a≤z<pl or q2c^+1-l≤z<c, there exists a dual pair of chains {Ck+1(1),Ck+1(2)} in {U1(k+1),U2(k+1)} such that |((PCk+1(1)-,Ck+1(2)-,QCk+1(1)-,Ck+1(2)-)(1)up-(PCk+1(1)-,Ck+1(2)-,QCk+1(1)-,Ck+1(2)-)(2)up)≤z|≥|(P1′-Q1′)≤z|. Assuming the claim and using the fact that T≤(PCk+1(1)-,Ck+1(2)-,QCk+1(1)-,Ck+1(2)-)up (which is because {U1,U2} is bounded by T,∅), we are done. The claim follows from Lemmas 41 and 42.
Case2. P1′ is obtained by adding a to P1 in position l from the left and adding d to P1 at the rightmost end (after p2c^); Q1′ is obtained from Q1 by adding b to the leftmost end of Q1 and adding c at the backward lth position of Q1. That is, P1′=P1∪˙{a,d}={p1<⋯<pl-1<a<pl<⋯<p2c^<d}, and Q1′=Q1∪˙{b,c}={b<q1<⋯<q2c^+1-l<c<q2c^+1-(l-1)<⋯<q2c^}.
Note thatpl-1<a<b<q1, a<b≤pl (since b>pl would require that a bumps pl in the bounded insertion process), and a<b<d<c. For z<a,
(18)|(T(1)-T(2))≤z|≥|(P1-Q1)≤z|=|(P1′-Q1′)≤z|,
where the first inequality follows from induction hypothesis and the second equality follows from the facts that pl-1<a<pl and a<b<q1. For z such that b≤z<d, note that
(19)|(P1′-Q1′)≤z|=|(P1′∪˙(ℕ∖Q1′))≤z|=|(P1′)≤z|+|(ℕ∖Q1′)≤z|=(|(P1)≤z|+1)+(|(ℕ∖Q1)≤z|-1)=|(P1)≤z|+|(ℕ∖Q1)≤z|=|(P1-Q1)≤z|.
Hence for b≤z<d, we have
(20)|(T(1)-T(2))≤z|≥|(P1-Q1)≤z|=|(P1′-Q1′)≤z|.
For z≥c,
(21)|(P1′-Q1′)≤z|=|(P1′∪˙(ℕ∖Q1′))≤z|=|(P1′)≤z|+|(ℕ∖Q1′)≤z|=(|(P1)≤z|+2)+(|(ℕ∖Q1)≤z|-2)=|(P1)≤z|+|(ℕ∖Q1)≤z|=|(P1-Q1)≤z|.
Hence for z≥c, we have
(22)|(T(1)-T(2))≤z|≥|(P1-Q1)≤z|=|(P1′-Q1′)≤z|.
It now remains to show that for z such that a≤z<b or d≤z<c,
(23)|(T(1)-T(2))≤z|≥|(P1′-Q1′)≤z|.
We claim that for z such that a≤z<b or d≤z<c, there exists a dual pair of chains {Sk+1(1),Sk+1(2)} in {U1(k+1),U2(k+1)} such that
(24)|({topmost  row  ofPSk+1(1),Sk+1(2)}∪˙(ℕ∖{topmost  row  ofQSk+1(1),Sk+1(2)}))≤z|≥|(P1′-Q1′)≤z|.
Assuming the claim, we are done as in Case 1. But the proof of the claim is similar to the proof of the claim made in Case 1, so we omit it.
Hence we are done with the required proof modulo, the proof of the claim made in Case 1. As mentioned in Case 1, the claim follows from Lemmas41 and 42.It now remains to prove Lemmas41 and 42. But before we go to prove these two lemmas, let us have a close look at their statements and try to understand the reason behind generating such complicated lemmas for the purpose of proving the claim. It is clear that once we understand the reason behind the statement of Lemma 41 being so complicated, Lemma 42 is a natural thing to be proved for finishing the proof of the claim. Observe that Lemma 41  looks like a Generalization of Lemma 11.1 (i) of [8] in the present case of the Orthogonal grassmannian. In Lemma 11.1 (i) of [8], Kreiman says that, for 1≤j≤m(k), there exists a chain Ck,j in U(k) which has j elements, the last of which has first component pj(k). We need to compare this statement of Kreiman with the statement of Lemma 41  here. The fact that we need a dual pair of chains here (in Lemma 41) instead of “a chain” (as mentioned in Lemma 11.1 (i) of [8]) is quite natural to believe. But it is not apparently clear as to why we need the dual pair of chains in Lemma 41 to satisfy a “strange” condition which is condition (##) (this condition is explained in Definition 40). We will now illustrate by an example (see Example 35 and the illustration associated to it below) that asking for the dual pair of chains in Lemma 41 to satisfy condition (##) is not at all strange, in fact it is the only natural thing to be asked for in the present case of the Orthogonal grassmannian. Another explanation for the same is provided in Remark 36.Example 35.
Letd=5 and β=(5,7,8,9,10). Consider the multiset U={(1,9),(3,7)} in ℕ2 contained in the underlying set of β¯×β. Applying the map BRSK (which is defined in Section 6 of [8]) to the multiset U, we get BRSK (U)=. Set C1,1=C2,1={(1,9)} and C2,2=U. It is easy to see that the chains C1,1,C2,1, and C2,2 satisfy the required conditions of lemma 11.1 (i) of [8].
On the other hand, let us take the pair of negative skew-symmetric lexicographic multisets onℕ2 given by U1,U2, where U1=U and U2=U#={(4,8),(2,10)}. Observe that OBRSK ({U,U#})=, where X=10 (we have denoted 10 by the alphabet X here because the young tableau package used here does not work with 2-digit integers). Set C1,1(1)={(1,9)},C1,1(2)={(2,10)}, C1,2(1)={(1,9)},C1,2(2)={(2,10)}, C2,1(1)={(1,9)},C2,1(2)={(2,10)}, C2,2(1)={(1,9)},C2,2(2)={(2,10)}, C2,3(1)=U1,C2,3(2)=U2, and C2,4(1)=U1,C2,4(2)=U2. It is easy to see that according to the notation of Lemma 41, we have m(k)=2 for k=1 and m(k)=4 for k=2. It can also be easily verified that, for each k∈{1,2} and each j∈{1,…,m(k)}, the dual pair {Ck,j(1),Ck,j(2)} of chains (as mentioned above) satisfies the required condition (which is condition (##) with respect to the bitableau (P(k),Q(k)) and the integer j) of Lemma 41. In fact, more is true; these dual pairs of chains satisfy condition (##)-strong (this condition is explained in Definition 39) with respect to the bitableau (P(k),Q(k)) and the integer j for every k∈{1,2} and j∈{1,…,m(k)}.Illustration Associated to Example35. A possible analog of Lemma 11.1 (i) of [8] in the present case of the Orthogonal grassmannian would be to demand for the dual pair of chains {Ck,j(1),Ck,j(2)} in Lemma 41  to have exactly j elements in Ck,j(1) (as well as in Ck,j(2)) and the last element of Ck,j(1) to have first coordinate pj(k). But it is easy to see from Example 35 that such a demand is meaningless. Instead, it would be natural and meaningful to demand for the dual pair {Ck,j(1),Ck,j(2)} of chains in Lemma 41  to satisfy condition (##)-strong (this is introduced in Definition 39) with respect to the bitableau (P(k),Q(k)) and the integer j. But in Lemma 41, we demand the dual pair of chains {Ck,j(1),Ck,j(2)} to satisfy condition (##) (this is introduced in Definition 40) which is little weaker a condition to be satisfied than condition (##)-strong. The reason behind this is that if we demand for the dual pair {Ck,j(1),Ck,j(2)} of chains to satisfy condition (##)-strong instead of condition (##), then the proof of Lemma 41  becomes more difficult. This difficulty (in the proof of Lemma 41) arises at the point(s) where we need to show that the dual pair {Ck+1,l(1),Ck+1,l(2)} of chains in {U1(k+1),U2(k+1)} satisfies condition (##)-strong. It is easier to show that the dual pair {Ck+1,l(1),Ck+1,l(2)} of chains satisfies condition (##). This completes the explanation behind the statement of Lemma 41  being so complicated.Remark 36.
Note that the properties of the dual pair{Ck,j(1),Ck,j(2)} of chains in Lemma 41  are a Generalization of the properties of the chain Ck,j of Lemma 11.1 of [8]. In fact, this Generalization precisely reveals the difference between the OBRSK algorithm (when applied to a dual pair of chains) in the present case of the Orthogonal Grassmannian and the BRSK algorithm of [8] (when applied to a chain) in the case of the ordinary Grassmannian. Equivalently, this Generalization reveals the difference between the concept of “𝔒-domination of a chain by an element of I(d)” in the case of the Orthogonal Grassmannian (as in Section 2.4.4 of [9]) and the concept of “domination of a chain by an element of I(d,2d)” in the case of the ordinary Grassmannian as in [6].
Let us now prove Lemmas41 and 42, but for the proofs we need some definitions which are the following.Definition 37.
Let(P,Q) and k1 be as in Definition 24. Let j∈{1,…,k1}. With notation as in Definition 24, one calls (p1j,q1j) the jth element of (P,Q)up, and we denote by (P,Q)≤jup the subset {(p11,q11),…,(p1j,q1j)} of (P,Q)up. See Example 38 for an illustration.Example 38.
Consider the row-strict notched bitableau(𝒫,𝒬) of Example 25. If we take j=3, then the jth element of (𝒫,𝒬)up is (T,K) and (𝒫,𝒬)≤jup equals {(H,D),(J,A),(T,K)}. We can work out similarly for other possible values of j.Definition 39.
Let(P,Q) be a negative skew-symmetric notched bitableau. Let {p1,…,p2c} and {q1,…,q2c} be the topmost row of P and Q, respectively, both listed in increasing order. Let j∈{1,…,2c}. We say that a dual pair {C1,C2} of chains in ℕ2 satisfies condition (##)-strong with respect to the bitableau (P,Q) and the integer j if all the following conditions are satisfied simultaneously.(i)
|C1|,|C2|≤j.(ii)
The first coordinate of thejth element of (PC1-,C2-,QC1-,C2-)up is pj.(iii)
All the entries which occur as first coordinates of elements ofC1 form a subset of the set of all entries which occur as first coordinates of elements of (PC1-,C2-,QC1-,C2-)≤jup.Definition 40.
Let(P,Q) be a negative skew-symmetric notched bitableau. Let {p1,…,p2c} and {q1,…,q2c} be the topmost row of P and Q, respectively, both listed in increasing order. Let j∈{1,…,2c}. One  says that a dual pair {C1,C2} of chains in ℕ2 satisfies condition (##) with respect to the bitableau (P,Q) and the integer j if all the following conditions are satisfied simultaneously.(i)
|C1|,|C2|≤j.(ii)
There exists an integerχj which is ≥j such that the first coordinate of the χjth element of (PC1-,C2-,QC1-,C2-)up is pj.(iii)
All the entries which occur as first coordinates of elements ofC1 form a subset of the set of all entries which occur as first coordinates of elements of (PC1-,C2-,QC1-,C2-)≤χjup.Lemma 41.
Let{U1={(a1,b1),…,(at,bt)},U2={(d1,c1),…,(dt,ct)}} be a pair of negative skew-symmetric lexicographic multisets on ℕ2. For k=1,…,t, let U1(k):={(a1,b1),…,(ak,bk)} and U2(k):={(dt+1-k,ct+1-k),…,(dt,ct)}. Let (P(k),Q(k)):=OBRSK({U1(k),U2(k)}). Define {p1(k),…,p2ck(k)} to be the topmost row of P(k) and {q1(k),…,q2ck(k)} the topmost row of Q(k), both listed in increasing order. Let m(k):=max{m∈{1,…,2ck}∣pm(k)<q1(k)}=|(P1(k))<q1(k)|. Then for 1≤j≤m(k), there exists a dual pair of chains {Ck,j(1),Ck,j(2)} in {U1(k),U2(k)} which satisfies condition (##) with respect to the bitableau (P(k),Q(k)) and the integer j. Let one denote by χk,j the integer corresponding to condition (##) (item number (ii), which is ≥j).Proof.
We prove this by induction onk, with k=1 the starting point of induction. When k=1, U1(1)={(a1,b1)} and U2(1)={(dt,ct)}. There are two possible cases, namely, when m(k)=1 and m(k)=2. In both the cases, for every j∈{1,…,m(k)}, take C1,j(1)={(a1,b1)} and C1,j(2)={(dt,ct)}. This proves the base case of induction.
Let(P,Q)=(P(k),Q(k)), a=ak+1, b=bk+1, c=ct-k, d=dt-k, (P′,Q′)=(P(k+1),Q(k+1)), {p1,…,p2c^}={p1(k),…,p2ck(k)}, and {q1,…,q2c^}={q1(k),…,q2ck(k)}. Note that {p1,…,p2c^}⊂{a1,…,ak}∪˙{dt+1-k,…,dt} and {q1,…,q2c^}⊂{b1,…,bk}∪˙{ct+1-k,…,ct}. Let P1 (resp., Q1) denote the topmost row of P (resp., Q). Similarly let P1′ (resp., Q1′) denote the topmost row of P′ (resp., Q′).
Sinceb is less than or equal to all elements of {b1,…,bk} and bi<ct+1-i∀i∈{1,…,t}, it follows that b≤ all elements of {b1,…,bk}∪˙{ct+1-k,…,ct}. Therefore a<b≤q1, and hence by duality c>d≥p2c^. We consider two cases corresponding to the two ways in which (P1′,Q1′) can be obtained from (P1,Q1).
Case1. P1′ is obtained by a bumping pl in P1, for some 1≤l≤2c^; that is, P1′=(P1∖{pl})∪˙{a} and Q1′=(Q1∖{q2c^+1-l})∪˙{c}.
The fact thata bumps pl implies that a≤pl and pl<b. Hence a≤pl<b≤q1, and therefore, by duality, we have c≥q2c^+1-l>d≥p2c^. This implies that m(k+1)=m(k). For j∈{1,…,m(k)}∖{l}, set Ck+1,j(1)=Ck,j(1) and Ck+1,j(2)=Ck,j(2) (Note that in these cases pj(k)=pj(k+1).). We now consider the case when j=l. If l=1, then set Ck+1,l(1)={(a,b)} and Ck+1,l(2)={(d,c)}. Otherwise consider the dual pair of chains {Ck,l-1(1),Ck,l-1(2)} in {U1(k),U2(k)}. Let Ck+1,l(1):=Ck,l-1(1)∪{(a,b)} and Ck+1,l(2):=Ck,l-1(2)∪{(d,c)}. Note that the dual pair {Ck+1,l(1),Ck+1,l(2)} of chains in {U1(k+1),U2(k+1)} satisfies the required conditions.
Case2. P1′ is obtained by adding a to P1 in position l from the left and adding d to P1 at the rightmost end (after p2c^); Q1′ is obtained from Q1 by adding b to the leftmost end of Q1 and adding c at the backward lth position of Q1. That is, P1′=P1∪˙{a,d}={p1<⋯<pl-1<a<pl<⋯<p2c^<d}, and Q1′∪˙{b,c}={b<q1<⋯<q2c^+1-l<c<q2c^+1-(l-1)<⋯<q2c^}.
Sincepl-1<a<b<q1, it follows that m(k)≥(l-1). Note that a<b≤pl (since b>pl would require that a bumps pl in the bounded insertion process). Thus m(k+1)=l. For j∈{1,…,l-1}, set Ck+1,j(1)=Ck,j(1) and Ck+1,j(2)=Ck,j(2). Consider the dual pair {Ck,l-1(1),Ck,l-1(2)} of chains in {U1(k),U2(k)}. Let Ck+1,l(1):=Ck,l-1(1)∪{(a,b)} and Ck+1,l(2):=Ck,l-1(2)∪{(d,c)}. Note that the dual pair {Ck+1,l(1),Ck+1,l(2)} of chains in {U1(k+1),U2(k+1)} satisfies the required conditions.Lemma 42.
Let(P,Q) be a negative skew-symmetric notched bitableau. Let a,b,c, and d be integers such that (P,Q)←b,ca,d is well defined. Let (P′,Q′):=(P,Q)←b,ca,d. Let P1 (resp., Q1) denote the topmost row of P (resp., Q). Similarly let P1′ (resp., Q1′) denote the topmost row of P′ (resp., Q′). Let {p1,…,p2c^} (resp., {q1,…,q2c^}) denote P1 (resp., Q1), both listed in increasing order. Suppose P1′ is obtained by a bumping pl in P1, for some 1≤l≤2c^; that is, suppose P1′=(P1∖{pl})∪˙{a} and Q1′=(Q1∖{q2c^+1-l})∪˙{c}. Let {C1,C2} be a dual pair of negative chains in ℕ2 which satisfies condition (##) with respect to the bitableau (P′,Q′) and the integer l. Let C1={(e1,f1),…,(er,fr),(a,b)} and C2={(d,c),(gr,hr),…,(g1,h1)}, where e1<⋯<er<a<pl<b<fr<⋯<f1 and h1>⋯>hr>c>q2c^+1-l>d>gr>⋯>g1. Then for z such that a≤z<pl or q2c^+1-l≤z<c, one has
(25)|((PC1,C2,QC1,C2)(1)up-(PC1,C2,QC1,C2)(2)up)≤z|≥|(P1′-Q1′)≤z|.Proof.
Note that fora≤z<pl, we have |((PC1,C2, QC1,C2)(1)up – (PC1,C2,QC1,C2)(2)up)≤z| = |({topmostrow of PC1,C2}-{topmostrow of QC1,C2})≤z| = |({topmostrow  of PC1,C2}∪˙(ℕ∖{topmostrowofQC1,C2}))≤z| = |{topmostrowofPC1,C2}≤z∪˙{ℕ}≤z|≥χl+z≥l+z, where the last equality (not inequality!) is because b≤ all entries in QC1,C2 and a≤z<pl<b (All the other inequalities and equalities being obvious.).
Also,p1<⋯<pl-1<a<pl<b≤q1<⋯<q2c^, and b<c. Thus for a≤z<pl, |(P1′-Q1′)≤z|=|(P1′∪˙(ℕ∖Q1′))≤z|=|(P1′∪˙ℕ)≤z|=l+z.
Hence we are done in the casea≤z<pl. Now for z such that q2c^+1-l≤z<c, we need to show that
(26)|{∪˙(ℕ∖{topmostrowofQC1,C2}))≤z}({topmostrowofPC1,C2}∪˙(ℕ∖{topmostrowofQC1,C2}))≤z|≥|(P1′∪˙(ℕ∖Q1′))≤z|.
Recall thatP1′=(P1∖{pl})∪˙{a} and Q1′=(Q1∖{q2c^+1-l})∪˙{c}. Since (P′,Q′)=(P,Q)←b,ca,d, therefore d≥ all entries of P′. Hence d≥ all entries of P1′. On the other hand, since a<pl<b, it follows from duality that c>q2c^+1-l>d. So we have p1<⋯<pl-1<a<pl+1<⋯<p2c^≤d<q2c^+1-l<c<q2c^+1-(l-1)<⋯<q2c^. It is now easy to observe that for z such that q2c^+1-l≤z<c, the number of elements in Q1′ which are ≤z is 2c^-l. Hence the number of elements in ℕ∖Q1′ which are ≤z will be z-(2c^-l).
It is also clear that all the elements ofP1′ are ≤z, and there are 2c^ many elements in P1′. Therefore,
(27)|(P1′∪˙(ℕ∖Q1′))≤z|=2c^+(z-(2c^-l))=2c^+z-2c^+l=z+l.
Letα1<⋯<α2c~ and β1<⋯<β2c~ denote the topmost rows of PC1,C2 and QC1,C2, respectively. It follows from the algorithm of OBRSK applied on {C1,C2} that d≥α2c^>⋯>α1. On the other hand, since a<pl<b, it follows from duality that c>q2c^+1-l>d. Hence combining all these, we have c>q2c^+1-l>d≥α2c~>α1. So for z such that q2c^+1-l≤z<c, the number of elements in the topmost row  of PC1,C2 which are ≤z is 2c~.
Since{C1,C2} satisfies condition (##) with respect to the bitableau (P′,Q′) and the integer l, it follows that there exists an integer χl(≥l) such that the χlth entry (counting from left to right) of the topmost row of PC1,C2 is a. Hence it follows from duality that the backward χlth entry (i.e., the χlth entry counting from right to left) of the topmost row of QC1,C2 is c. Therefore for z such that q2c^+1-l≤z<c, the number of elements in the topmost row of QC1,C2 which are ≤z is equal to X0, where X0 is some nonnegative integer such that X0≤2c~-χl. But χl≥l, hence -χl≤-l, and therefore X0≤2c~-χl≤2c~-l.
Therefore, the number of elements in(ℕ∖{topmostrowofQC1,C2}) which are ≤z is z-X0. Hence |({topmostrowofPC1,C2}∪˙(ℕ∖{topmostrowofQC1,C2}))≤z| = 2c~+z-X0≥2c~+z-(2c~-l)=z+l=|(P1′∪˙(ℕ∖Q1′))≤z|.


## 9.1. Proof of Lemma16

Proof.
The proof is by induction. The base case of induction is easy to see, and hence we omit the details. Let{π1(t-1),π2(t-1)} be a pair of negative skew-symmetric lexicographic multisets on ℕ2, where π1(t-1)={(a1,b1),…,(at-1,bt-1)} and π2(t-1)={(d2,c2),…,(dt,ct)}. Let (P,Q)=OBRSK({π1(t-1),π2(t-1)}). Assume inductively that (P,Q) is a negative skew-symmetric notched bitableau. Now let {π1(t),π2(t)} be a pair of negative skew-symmetric lexicographic multisets on ℕ2 such that π1(t)=π1 and π2(t)=π2, where π1 and π2 are given by π1={(a1,b1),…,(at,bt)} and π2={(d1,c1),…,(dt,ct)}. Let (P′,Q′)=OBRSK({π1(t),π2(t)}). It suffices to show that (P′,Q′) is also a negative skew-symmetric notched bitableau.
We will first prove that(P′,Q′) is a negative semistandard notched bitableau. The fact that P′ is row strict follows from Lemma 15. It then follows from duality that Q′ is also row strict. Hence (P′,Q′) is row strict. Let r′ be the total number of rows of P′(orQ′). Let Pi′,Qi′,Pi, and Qi denote the multiset consisting of all elements in the ith row of P′,Q′,P, and Q, respectively. In fact, these multisets (Pi′,Qi′,Pi,Qi) happen to be sets because P′,Q′,P, and Q are row strict. It needs to be shown that Pi′-Qi′≤Pi+1′-Qi+1′ for 1≤i≤r′-1, and Pr′′-Qr′′⋖∅ (see Section 4 of [8] for the notation ⋖). We will prove the former statement first. To prove that Pi′-Qi′≤Pi+1′-Qi+1′ for 1≤i≤r′-1, there will be three nontrivial possibilities which will be registered below as Cases 1, 2, and 3, respectively.
Case1. at and d1 are added to the first row of P, and c1 and bt are added to the first row of Q. All rows other than the 1st row of P and Q remain unchanged.
In this case, it suffices to show thatP1∪˙Q2∪˙{at,d1}≤P2∪˙Q1∪˙{c1,bt}. It follows from induction hypothesis that P1∪˙Q2≤P2∪˙Q1. Since {π1(t),π2(t)} is a pair of negative skew-symmetric lexicographic multisets on ℕ2, we have at<bt and d1<c1. Therefore at≤bt and d1≤c1. These facts together with Lemma 34 (applied twice) imply easily that P1∪˙Q2∪˙{at,d1}≤P2∪˙Q1∪˙{c1,bt}. Hence we are done in this case.
Case2.xp bumps yp from Pi, yp bumps zp from Pi+1, and the dual bumping happens on Qi and Qi+1. Let us express the dual bumping by saying that xq bumps yq from Qi and yq bumps zq from Qi+1.
Clearly then,xp≤yp≤zp<bt, and hence xq≥yq≥zq by the property (ii) in the definition of a skew-symmetric notched bitableau. Again since all entries in Q are ≥bt, we get that xq≥yq≥zq≥bt.
We need to show thatPi′-Qi′≤Pi+1′-Qi+1′, or, in other words, Pi′∪˙Qi+1′≤Pi+1′∪˙Qi′. Note that Pi′∪˙Qi+1′=[(Pi∪˙Qi+1)∖{yp,zq}]∪˙{xp,yq} and Pi+1′∪˙Qi′=[(Pi+1∪˙Qi)∖{zp,yq}]∪˙{xq,yp}. It suffices to show that (Pi∪˙Qi+1)∖{yp,zq}≤(Pi+1∪˙Qi)∖{zp,yq} because if we can prove this much, then since xp≤yp and xq≥yq we can apply Lemma 34 and get done with the proof as in Case 1.
We will now prove that(Pi∪˙Qi+1)∖{yp,zq}≤(Pi+1∪˙Qi)∖{zp,yq}. Since zp<bt and all entries of Qi are ≥bt, therefore zp is the smallest element in Pi+1∪˙Qi which is ≥yp. It then follows from duality that zq is the biggest element in Pi∪˙Qi+1, that is, ≤yq. Also since yp<bt≤zq, we have yp<zq, and hence by duality yq>zp. Also by induction hypothesis, Pi∪˙Qi+1≤Pi+1∪˙Qi. All these facts put together prove the required thing easily, and we are done in this case.
Case3. xp bumps yp from Pi, and yp along with d1 are added to Pi+1. The dual phenomenon happens with Qi and Qi+1; we express the dual phenomenon by saying that xq bumps yq from Qi, and yq along with bt are added to Qi+1.
Clearly then,xp≤yp<bt, and hence xq≥yq by the property (ii) in the definition of a Skew-symmetric notched bitableau. Again since all entries in Q are ≥bt, we get that xq≥yq≥bt. Also yp<d1 since the position of d1 is at the rightmost end of that row of Pi+1 in which yp is placed. Hence by duality, we have yq>bt.
We need to show thatPi′-Qi′≤Pi+1′-Qi+1′, or, in other words, Pi′∪˙Qi+1′≤Pi+1′∪˙Qi′. Note that Pi′∪˙Qi+1′=(Pi∪˙Qi+1∖{yp})∪˙{bt}∪˙{xp,yq}, and Pi+1′∪˙Qi′=(Pi+1∪˙Qi∖{yq})∪˙{d1}∪˙{xq,yp}. Using Lemma 34, it is enough to prove that (Pi∪˙Qi+1∖{yp})∪˙{bt}≤(Pi+1∪˙Qi∖{yq})∪˙{d1}, which we will do now.
LetPi∪˙Qi+1 be given by X1≤⋯≤X2n, and let Pi+1∪˙Qi be given by Y1≤⋯≤Y2n (since the bitableau (P,Q) is of even size, the cardinality of Pi∪˙Qi+1 or of Pi+1∪˙Qi is even). Note that there are no elements of Pi+1 which are ≥yp and <bt. Also since bt is ≤ all elements of Qi, therefore we can conclude that there are no elements of Pi+1∪˙Qi which are ≥yp and <bt. It follows from duality that there are no elements of Pi∪˙Qi+1 which are ≤yq as well as >d1. Let (Pi∪˙Qi+1∖{yp})∪˙{bt} be given by
(15)X1≤⋯≤Xi-1≤Xi+1≤⋯≤Xj≤bt≤Xj+1≤⋯≤X2n
and let (Pi+1∪˙Qi∖{yq})∪˙{d1} be given by
(16)Y1≤⋯≤Y2n-j≤d1≤Y2n-j+1≤⋯≤Y2n-i≤Y2n-i+2≤⋯≤Y2n.
This means that yp=Xi and its dual yq=Y2n-i+1 (the ith element counting from right to left in the ordered sequence Y1≤⋯≤Y2n). Also since yp=Xi and yp<bt, it follows that i<j. Observe that bt is the jth element (counting from left to right) in the ordered sequence X1≤⋯≤Xi-1≤Xi+1≤⋯≤Xj≤bt≤Xj+1≤⋯≤X2n of integers and d1 (the dual of bt with respect to the pair {π1,π2} of multisets) is the jth element (counting from right to left) in the ordered sequence Y1≤⋯≤Y2n-j≤d1≤Y2n-j+1≤⋯≤Y2n-i≤Y2n-i+2≤⋯≤Y2n of integers. Consider any element in (Pi∪˙Qi+1∖{yp})∪˙{bt}. It can either be Xk for some k≠i or bt. If it is Xk for some k≠i, then we either need to show that Xk≤Yk, Xk≤d1, or Xk≤Yk-1 (these are the only 3 possibilities; the last possibility Xk≤Yk-1 happens if and only if k∈{2n-j+2,…,2n-i+1}). If it is bt, then we need to show that bt≤ the jth element (counting from left to right) of (Pi+1∪˙Qi∖{yq})∪˙{d1}, which can either be Yj, d1, or Yj-1.
After all these observation(s), the proof follows easily using the inequalitiesi<j, xp≤yp<bt, xq≥yq≥bt, yp<d1, and yq>bt and the facts that there are no elements of Pi+1∪˙Qi which lie in the interval [yp,bt) and that there are no elements of Pi∪˙Qi+1 which lie in the interval (d1,yq].
We have now proved that(P′,Q′) is a semistandard notched bitableau. To prove that the semistandard notched bitableau (P′,Q′) is negative, it is enough to prove that Pr′′-Qr′′⋖∅, where r′ denotes the total number of rows of P′(orQ′). If Pr′′=Pr′ and Qr′′=Qr′, then since (P,Q) is assumed to be a negative skew-symmetric notched bitableau, it follows immediately that Pr′′-Qr′′⋖∅. Otherwise, we do the following: let Pr′′ and Qr′′ be given by λ1<⋯<λs′ and δ1<⋯<δs′, respectively. It follows easily from the way the OBRSK algorithm works that there exists at least one entry in Pr′′ which is <bt. Let λl denote the largest entry in Pr′′ which is <bt. Since bt≤δ1, it now follows easily that λj<δj∀j∈{1,…,l}. Since λl is the largest entry in Pr′′ which is <bt, therefore l≥1, and it follows from duality that δs′+1-l=δs′-(l-1)>d1. Hence δs′≥δs′-(l-1)>d1=λs′. It only remains to show that if l+1<s′, then λj<δj∀j∈{l+1,…,s′-1}. Clearly since l≥1 and l+1<s′, it follows that, in this case, the last row of P (resp., Q) is the r′th row, namely Pr′ (resp., Qr′), and δs′-(l-1) is the new box of the dual insertion in Qr′. Since λs′=d1,δs′-(l-1)>d1, Pr′-Qr′⋖∅ by induction hypothesis and λj<λs′∀j∈{l+1,…,s′-1}, it is now easy to see that λj<δj∀j∈{l+1,…,s′-1}.
Hence we have proved that(P′,Q′) is a negative semistandard notched bitableau. The fact that (P′,Q′) is skew-symmetric is easy to see from the very construction of the OBRSK algorithm and from the fact that the pair {π1(t),π2(t)} of multisets is skew-symmetric, lexicographic.


## 9.2. Proof of Lemma27

Proof (proof of Lemma27).
LetT be a negative subset of ℕ2 such that T(1),T(2) are subsets of ℕ. Suppose that {U1={(a1,b1),…,(at,bt)},U2={(d1,c1),…,(dt,ct)}} is a pair of negative skew-symmetric lexicographic multisets on ℕ2. For each m=1,…,t, let U1(m):={(a1,b1),…,(am,bm)} and U2(m):={(dt+1-m,ct+1-m),…,(dt,ct)}. Let (P(m),Q(m)):=OBRSK({U1(m),U2(m)}). For proving Lemma 27, it suffices to show that if {U1,U2} is bounded by T,∅, then (P(m),Q(m)) is bounded by T,∅ for all m=1,…,t because then the proof of Lemma 27 will follow similarly as the proofs of parts (iii), (iv), and (v) of  Lemma  11.1 in [8].
Assume that{U1,U2} is bounded by T,∅. For every m∈{1,…,t}, let P1(m) (resp., Q1(m)) denote the topmost row of P(m) (resp., Q(m)). We need to show that T(1)-T(2)≤P1(m)-Q1(m). Equivalently, we need to show that for all positive integers z, |(T(1)-T(2))≤z|≥|(P1(m)-Q1(m))≤z|, where we use the definition A-B:=A∪˙(ℕ∖B) for any two subsets A and B of ℕ (see Section 4 of [8]). We will prove this by induction on m. Let us first see the base case of induction. When m=1, U1(m)={(a1,b1)} and U2(m)={(dt,ct)}. Let C1={(a1,b1)} and C2={(dt,ct)}. Since {U1,U2} is bounded by T,∅, it follows that for the dual pair {C1,C2} of chains in {U1,U2}, we have T≤(PC1-,C2-,QC1-,C2-)up≤∅. But it can be easily seen that (PC1-,C2-,QC1-,C2-)up={(a1,b1),(dt,ct)}. This proves the base case of induction. Fix an integer k∈{1,…,t-1}, and assume that (P(m),Q(m)) is bounded by T,∅ for all m≤k. We will now show that (P(k+1),Q(k+1)) is bounded by T,∅.
Before we begin the proof, let us fix some notation for the rest of this subsection. Define{p1(k),…,p2ck(k)} to be the topmost row of P(k) and {q1(k),…,q2ck(k)} the topmost row of Q(k), both listed in increasing order. Let (P,Q)=(P(k),Q(k)), a=ak+1, b=bk+1, c=ct-k, d=dt-k, (P′,Q′)=(P(k+1),Q(k+1)), {p1,…,p2c^}={p1(k),…,p2ck(k)}, and {q1,…,q2c^}={q1(k),…,q2ck(k)}. Note that {p1,…,p2c^}⊂{a1,…,ak}∪˙{dt+1-k,…,dt} and {q1,…,q2c^}⊂{b1,…,bk}∪˙{ct+1-k,…,ct}. Let P1 (resp., Q1) denote the topmost row of P (resp., Q). Similarly let P1′ (resp., Q1′) denote the topmost row of P′ (resp., Q′). We consider two cases corresponding to the two ways in which (P1′,Q1′) can be obtained from (P1,Q1).
Case1. P1′ is obtained by a bumping pl in P1, for some 1≤l≤2c^, that is, P1′=(P1∖{pl})∪˙{a} and Q1′=(Q1∖{q2c^+1-l})∪˙{c}.
Sinceb is less than or equal to all elements of {b1,…,bk} and bi<ct+1-i∀i∈{1,…,t}, it follows that b≤ all elements of {b1,…,bk}∪˙{ct+1-k,…,ct}. Therefore a<b≤q1, and hence by duality c>d≥p2c^. The fact that a bumps pl implies that a≤pl and pl<b. Hence a≤pl<b≤q1, and therefore by duality, we have c≥q2c^+1-l>d≥p2c^. For z<a, pl≤z<q2c^+1-l, or z≥c,
(17)|(T(1)-T(2))≤z|≥|(P1-Q1)≤z|=|(P1′-Q1′)≤z|,
where the first inequality follows from induction hypothesis, and second equality follows from the facts that pl-1<a≤pl<b≤q1≤q2c^+1-l≤c and p2c^≤d<c. If a=pl, then we are done. Thus we assume that a<pl (and hence by duality that c>q2c^+1-l). We now need to consider only two possible positions of z, namely, a≤z<pl and q2c^+1-l≤z<c. We claim that, for z such that a≤z<pl or q2c^+1-l≤z<c, there exists a dual pair of chains {Ck+1(1),Ck+1(2)} in {U1(k+1),U2(k+1)} such that |((PCk+1(1)-,Ck+1(2)-,QCk+1(1)-,Ck+1(2)-)(1)up-(PCk+1(1)-,Ck+1(2)-,QCk+1(1)-,Ck+1(2)-)(2)up)≤z|≥|(P1′-Q1′)≤z|. Assuming the claim and using the fact that T≤(PCk+1(1)-,Ck+1(2)-,QCk+1(1)-,Ck+1(2)-)up (which is because {U1,U2} is bounded by T,∅), we are done. The claim follows from Lemmas 41 and 42.
Case2. P1′ is obtained by adding a to P1 in position l from the left and adding d to P1 at the rightmost end (after p2c^); Q1′ is obtained from Q1 by adding b to the leftmost end of Q1 and adding c at the backward lth position of Q1. That is, P1′=P1∪˙{a,d}={p1<⋯<pl-1<a<pl<⋯<p2c^<d}, and Q1′=Q1∪˙{b,c}={b<q1<⋯<q2c^+1-l<c<q2c^+1-(l-1)<⋯<q2c^}.
Note thatpl-1<a<b<q1, a<b≤pl (since b>pl would require that a bumps pl in the bounded insertion process), and a<b<d<c. For z<a,
(18)|(T(1)-T(2))≤z|≥|(P1-Q1)≤z|=|(P1′-Q1′)≤z|,
where the first inequality follows from induction hypothesis and the second equality follows from the facts that pl-1<a<pl and a<b<q1. For z such that b≤z<d, note that
(19)|(P1′-Q1′)≤z|=|(P1′∪˙(ℕ∖Q1′))≤z|=|(P1′)≤z|+|(ℕ∖Q1′)≤z|=(|(P1)≤z|+1)+(|(ℕ∖Q1)≤z|-1)=|(P1)≤z|+|(ℕ∖Q1)≤z|=|(P1-Q1)≤z|.
Hence for b≤z<d, we have
(20)|(T(1)-T(2))≤z|≥|(P1-Q1)≤z|=|(P1′-Q1′)≤z|.
For z≥c,
(21)|(P1′-Q1′)≤z|=|(P1′∪˙(ℕ∖Q1′))≤z|=|(P1′)≤z|+|(ℕ∖Q1′)≤z|=(|(P1)≤z|+2)+(|(ℕ∖Q1)≤z|-2)=|(P1)≤z|+|(ℕ∖Q1)≤z|=|(P1-Q1)≤z|.
Hence for z≥c, we have
(22)|(T(1)-T(2))≤z|≥|(P1-Q1)≤z|=|(P1′-Q1′)≤z|.
It now remains to show that for z such that a≤z<b or d≤z<c,
(23)|(T(1)-T(2))≤z|≥|(P1′-Q1′)≤z|.
We claim that for z such that a≤z<b or d≤z<c, there exists a dual pair of chains {Sk+1(1),Sk+1(2)} in {U1(k+1),U2(k+1)} such that
(24)|({topmost  row  ofPSk+1(1),Sk+1(2)}∪˙(ℕ∖{topmost  row  ofQSk+1(1),Sk+1(2)}))≤z|≥|(P1′-Q1′)≤z|.
Assuming the claim, we are done as in Case 1. But the proof of the claim is similar to the proof of the claim made in Case 1, so we omit it.
Hence we are done with the required proof modulo, the proof of the claim made in Case 1. As mentioned in Case 1, the claim follows from Lemmas41 and 42.It now remains to prove Lemmas41 and 42. But before we go to prove these two lemmas, let us have a close look at their statements and try to understand the reason behind generating such complicated lemmas for the purpose of proving the claim. It is clear that once we understand the reason behind the statement of Lemma 41 being so complicated, Lemma 42 is a natural thing to be proved for finishing the proof of the claim. Observe that Lemma 41  looks like a Generalization of Lemma 11.1 (i) of [8] in the present case of the Orthogonal grassmannian. In Lemma 11.1 (i) of [8], Kreiman says that, for 1≤j≤m(k), there exists a chain Ck,j in U(k) which has j elements, the last of which has first component pj(k). We need to compare this statement of Kreiman with the statement of Lemma 41  here. The fact that we need a dual pair of chains here (in Lemma 41) instead of “a chain” (as mentioned in Lemma 11.1 (i) of [8]) is quite natural to believe. But it is not apparently clear as to why we need the dual pair of chains in Lemma 41 to satisfy a “strange” condition which is condition (##) (this condition is explained in Definition 40). We will now illustrate by an example (see Example 35 and the illustration associated to it below) that asking for the dual pair of chains in Lemma 41 to satisfy condition (##) is not at all strange, in fact it is the only natural thing to be asked for in the present case of the Orthogonal grassmannian. Another explanation for the same is provided in Remark 36.Example 35.
Letd=5 and β=(5,7,8,9,10). Consider the multiset U={(1,9),(3,7)} in ℕ2 contained in the underlying set of β¯×β. Applying the map BRSK (which is defined in Section 6 of [8]) to the multiset U, we get BRSK (U)=. Set C1,1=C2,1={(1,9)} and C2,2=U. It is easy to see that the chains C1,1,C2,1, and C2,2 satisfy the required conditions of lemma 11.1 (i) of [8].
On the other hand, let us take the pair of negative skew-symmetric lexicographic multisets onℕ2 given by U1,U2, where U1=U and U2=U#={(4,8),(2,10)}. Observe that OBRSK ({U,U#})=, where X=10 (we have denoted 10 by the alphabet X here because the young tableau package used here does not work with 2-digit integers). Set C1,1(1)={(1,9)},C1,1(2)={(2,10)}, C1,2(1)={(1,9)},C1,2(2)={(2,10)}, C2,1(1)={(1,9)},C2,1(2)={(2,10)}, C2,2(1)={(1,9)},C2,2(2)={(2,10)}, C2,3(1)=U1,C2,3(2)=U2, and C2,4(1)=U1,C2,4(2)=U2. It is easy to see that according to the notation of Lemma 41, we have m(k)=2 for k=1 and m(k)=4 for k=2. It can also be easily verified that, for each k∈{1,2} and each j∈{1,…,m(k)}, the dual pair {Ck,j(1),Ck,j(2)} of chains (as mentioned above) satisfies the required condition (which is condition (##) with respect to the bitableau (P(k),Q(k)) and the integer j) of Lemma 41. In fact, more is true; these dual pairs of chains satisfy condition (##)-strong (this condition is explained in Definition 39) with respect to the bitableau (P(k),Q(k)) and the integer j for every k∈{1,2} and j∈{1,…,m(k)}.Illustration Associated to Example35. A possible analog of Lemma 11.1 (i) of [8] in the present case of the Orthogonal grassmannian would be to demand for the dual pair of chains {Ck,j(1),Ck,j(2)} in Lemma 41  to have exactly j elements in Ck,j(1) (as well as in Ck,j(2)) and the last element of Ck,j(1) to have first coordinate pj(k). But it is easy to see from Example 35 that such a demand is meaningless. Instead, it would be natural and meaningful to demand for the dual pair {Ck,j(1),Ck,j(2)} of chains in Lemma 41  to satisfy condition (##)-strong (this is introduced in Definition 39) with respect to the bitableau (P(k),Q(k)) and the integer j. But in Lemma 41, we demand the dual pair of chains {Ck,j(1),Ck,j(2)} to satisfy condition (##) (this is introduced in Definition 40) which is little weaker a condition to be satisfied than condition (##)-strong. The reason behind this is that if we demand for the dual pair {Ck,j(1),Ck,j(2)} of chains to satisfy condition (##)-strong instead of condition (##), then the proof of Lemma 41  becomes more difficult. This difficulty (in the proof of Lemma 41) arises at the point(s) where we need to show that the dual pair {Ck+1,l(1),Ck+1,l(2)} of chains in {U1(k+1),U2(k+1)} satisfies condition (##)-strong. It is easier to show that the dual pair {Ck+1,l(1),Ck+1,l(2)} of chains satisfies condition (##). This completes the explanation behind the statement of Lemma 41  being so complicated.Remark 36.
Note that the properties of the dual pair{Ck,j(1),Ck,j(2)} of chains in Lemma 41  are a Generalization of the properties of the chain Ck,j of Lemma 11.1 of [8]. In fact, this Generalization precisely reveals the difference between the OBRSK algorithm (when applied to a dual pair of chains) in the present case of the Orthogonal Grassmannian and the BRSK algorithm of [8] (when applied to a chain) in the case of the ordinary Grassmannian. Equivalently, this Generalization reveals the difference between the concept of “𝔒-domination of a chain by an element of I(d)” in the case of the Orthogonal Grassmannian (as in Section 2.4.4 of [9]) and the concept of “domination of a chain by an element of I(d,2d)” in the case of the ordinary Grassmannian as in [6].
Let us now prove Lemmas41 and 42, but for the proofs we need some definitions which are the following.Definition 37.
Let(P,Q) and k1 be as in Definition 24. Let j∈{1,…,k1}. With notation as in Definition 24, one calls (p1j,q1j) the jth element of (P,Q)up, and we denote by (P,Q)≤jup the subset {(p11,q11),…,(p1j,q1j)} of (P,Q)up. See Example 38 for an illustration.Example 38.
Consider the row-strict notched bitableau(𝒫,𝒬) of Example 25. If we take j=3, then the jth element of (𝒫,𝒬)up is (T,K) and (𝒫,𝒬)≤jup equals {(H,D),(J,A),(T,K)}. We can work out similarly for other possible values of j.Definition 39.
Let(P,Q) be a negative skew-symmetric notched bitableau. Let {p1,…,p2c} and {q1,…,q2c} be the topmost row of P and Q, respectively, both listed in increasing order. Let j∈{1,…,2c}. We say that a dual pair {C1,C2} of chains in ℕ2 satisfies condition (##)-strong with respect to the bitableau (P,Q) and the integer j if all the following conditions are satisfied simultaneously.(i)
|C1|,|C2|≤j.(ii)
The first coordinate of thejth element of (PC1-,C2-,QC1-,C2-)up is pj.(iii)
All the entries which occur as first coordinates of elements ofC1 form a subset of the set of all entries which occur as first coordinates of elements of (PC1-,C2-,QC1-,C2-)≤jup.Definition 40.
Let(P,Q) be a negative skew-symmetric notched bitableau. Let {p1,…,p2c} and {q1,…,q2c} be the topmost row of P and Q, respectively, both listed in increasing order. Let j∈{1,…,2c}. One  says that a dual pair {C1,C2} of chains in ℕ2 satisfies condition (##) with respect to the bitableau (P,Q) and the integer j if all the following conditions are satisfied simultaneously.(i)
|C1|,|C2|≤j.(ii)
There exists an integerχj which is ≥j such that the first coordinate of the χjth element of (PC1-,C2-,QC1-,C2-)up is pj.(iii)
All the entries which occur as first coordinates of elements ofC1 form a subset of the set of all entries which occur as first coordinates of elements of (PC1-,C2-,QC1-,C2-)≤χjup.Lemma 41.
Let{U1={(a1,b1),…,(at,bt)},U2={(d1,c1),…,(dt,ct)}} be a pair of negative skew-symmetric lexicographic multisets on ℕ2. For k=1,…,t, let U1(k):={(a1,b1),…,(ak,bk)} and U2(k):={(dt+1-k,ct+1-k),…,(dt,ct)}. Let (P(k),Q(k)):=OBRSK({U1(k),U2(k)}). Define {p1(k),…,p2ck(k)} to be the topmost row of P(k) and {q1(k),…,q2ck(k)} the topmost row of Q(k), both listed in increasing order. Let m(k):=max{m∈{1,…,2ck}∣pm(k)<q1(k)}=|(P1(k))<q1(k)|. Then for 1≤j≤m(k), there exists a dual pair of chains {Ck,j(1),Ck,j(2)} in {U1(k),U2(k)} which satisfies condition (##) with respect to the bitableau (P(k),Q(k)) and the integer j. Let one denote by χk,j the integer corresponding to condition (##) (item number (ii), which is ≥j).Proof.
We prove this by induction onk, with k=1 the starting point of induction. When k=1, U1(1)={(a1,b1)} and U2(1)={(dt,ct)}. There are two possible cases, namely, when m(k)=1 and m(k)=2. In both the cases, for every j∈{1,…,m(k)}, take C1,j(1)={(a1,b1)} and C1,j(2)={(dt,ct)}. This proves the base case of induction.
Let(P,Q)=(P(k),Q(k)), a=ak+1, b=bk+1, c=ct-k, d=dt-k, (P′,Q′)=(P(k+1),Q(k+1)), {p1,…,p2c^}={p1(k),…,p2ck(k)}, and {q1,…,q2c^}={q1(k),…,q2ck(k)}. Note that {p1,…,p2c^}⊂{a1,…,ak}∪˙{dt+1-k,…,dt} and {q1,…,q2c^}⊂{b1,…,bk}∪˙{ct+1-k,…,ct}. Let P1 (resp., Q1) denote the topmost row of P (resp., Q). Similarly let P1′ (resp., Q1′) denote the topmost row of P′ (resp., Q′).
Sinceb is less than or equal to all elements of {b1,…,bk} and bi<ct+1-i∀i∈{1,…,t}, it follows that b≤ all elements of {b1,…,bk}∪˙{ct+1-k,…,ct}. Therefore a<b≤q1, and hence by duality c>d≥p2c^. We consider two cases corresponding to the two ways in which (P1′,Q1′) can be obtained from (P1,Q1).
Case1. P1′ is obtained by a bumping pl in P1, for some 1≤l≤2c^; that is, P1′=(P1∖{pl})∪˙{a} and Q1′=(Q1∖{q2c^+1-l})∪˙{c}.
The fact thata bumps pl implies that a≤pl and pl<b. Hence a≤pl<b≤q1, and therefore, by duality, we have c≥q2c^+1-l>d≥p2c^. This implies that m(k+1)=m(k). For j∈{1,…,m(k)}∖{l}, set Ck+1,j(1)=Ck,j(1) and Ck+1,j(2)=Ck,j(2) (Note that in these cases pj(k)=pj(k+1).). We now consider the case when j=l. If l=1, then set Ck+1,l(1)={(a,b)} and Ck+1,l(2)={(d,c)}. Otherwise consider the dual pair of chains {Ck,l-1(1),Ck,l-1(2)} in {U1(k),U2(k)}. Let Ck+1,l(1):=Ck,l-1(1)∪{(a,b)} and Ck+1,l(2):=Ck,l-1(2)∪{(d,c)}. Note that the dual pair {Ck+1,l(1),Ck+1,l(2)} of chains in {U1(k+1),U2(k+1)} satisfies the required conditions.
Case2. P1′ is obtained by adding a to P1 in position l from the left and adding d to P1 at the rightmost end (after p2c^); Q1′ is obtained from Q1 by adding b to the leftmost end of Q1 and adding c at the backward lth position of Q1. That is, P1′=P1∪˙{a,d}={p1<⋯<pl-1<a<pl<⋯<p2c^<d}, and Q1′∪˙{b,c}={b<q1<⋯<q2c^+1-l<c<q2c^+1-(l-1)<⋯<q2c^}.
Sincepl-1<a<b<q1, it follows that m(k)≥(l-1). Note that a<b≤pl (since b>pl would require that a bumps pl in the bounded insertion process). Thus m(k+1)=l. For j∈{1,…,l-1}, set Ck+1,j(1)=Ck,j(1) and Ck+1,j(2)=Ck,j(2). Consider the dual pair {Ck,l-1(1),Ck,l-1(2)} of chains in {U1(k),U2(k)}. Let Ck+1,l(1):=Ck,l-1(1)∪{(a,b)} and Ck+1,l(2):=Ck,l-1(2)∪{(d,c)}. Note that the dual pair {Ck+1,l(1),Ck+1,l(2)} of chains in {U1(k+1),U2(k+1)} satisfies the required conditions.Lemma 42.
Let(P,Q) be a negative skew-symmetric notched bitableau. Let a,b,c, and d be integers such that (P,Q)←b,ca,d is well defined. Let (P′,Q′):=(P,Q)←b,ca,d. Let P1 (resp., Q1) denote the topmost row of P (resp., Q). Similarly let P1′ (resp., Q1′) denote the topmost row of P′ (resp., Q′). Let {p1,…,p2c^} (resp., {q1,…,q2c^}) denote P1 (resp., Q1), both listed in increasing order. Suppose P1′ is obtained by a bumping pl in P1, for some 1≤l≤2c^; that is, suppose P1′=(P1∖{pl})∪˙{a} and Q1′=(Q1∖{q2c^+1-l})∪˙{c}. Let {C1,C2} be a dual pair of negative chains in ℕ2 which satisfies condition (##) with respect to the bitableau (P′,Q′) and the integer l. Let C1={(e1,f1),…,(er,fr),(a,b)} and C2={(d,c),(gr,hr),…,(g1,h1)}, where e1<⋯<er<a<pl<b<fr<⋯<f1 and h1>⋯>hr>c>q2c^+1-l>d>gr>⋯>g1. Then for z such that a≤z<pl or q2c^+1-l≤z<c, one has
(25)|((PC1,C2,QC1,C2)(1)up-(PC1,C2,QC1,C2)(2)up)≤z|≥|(P1′-Q1′)≤z|.Proof.
Note that fora≤z<pl, we have |((PC1,C2, QC1,C2)(1)up – (PC1,C2,QC1,C2)(2)up)≤z| = |({topmostrow of PC1,C2}-{topmostrow of QC1,C2})≤z| = |({topmostrow  of PC1,C2}∪˙(ℕ∖{topmostrowofQC1,C2}))≤z| = |{topmostrowofPC1,C2}≤z∪˙{ℕ}≤z|≥χl+z≥l+z, where the last equality (not inequality!) is because b≤ all entries in QC1,C2 and a≤z<pl<b (All the other inequalities and equalities being obvious.).
Also,p1<⋯<pl-1<a<pl<b≤q1<⋯<q2c^, and b<c. Thus for a≤z<pl, |(P1′-Q1′)≤z|=|(P1′∪˙(ℕ∖Q1′))≤z|=|(P1′∪˙ℕ)≤z|=l+z.
Hence we are done in the casea≤z<pl. Now for z such that q2c^+1-l≤z<c, we need to show that
(26)|{∪˙(ℕ∖{topmostrowofQC1,C2}))≤z}({topmostrowofPC1,C2}∪˙(ℕ∖{topmostrowofQC1,C2}))≤z|≥|(P1′∪˙(ℕ∖Q1′))≤z|.
Recall thatP1′=(P1∖{pl})∪˙{a} and Q1′=(Q1∖{q2c^+1-l})∪˙{c}. Since (P′,Q′)=(P,Q)←b,ca,d, therefore d≥ all entries of P′. Hence d≥ all entries of P1′. On the other hand, since a<pl<b, it follows from duality that c>q2c^+1-l>d. So we have p1<⋯<pl-1<a<pl+1<⋯<p2c^≤d<q2c^+1-l<c<q2c^+1-(l-1)<⋯<q2c^. It is now easy to observe that for z such that q2c^+1-l≤z<c, the number of elements in Q1′ which are ≤z is 2c^-l. Hence the number of elements in ℕ∖Q1′ which are ≤z will be z-(2c^-l).
It is also clear that all the elements ofP1′ are ≤z, and there are 2c^ many elements in P1′. Therefore,
(27)|(P1′∪˙(ℕ∖Q1′))≤z|=2c^+(z-(2c^-l))=2c^+z-2c^+l=z+l.
Letα1<⋯<α2c~ and β1<⋯<β2c~ denote the topmost rows of PC1,C2 and QC1,C2, respectively. It follows from the algorithm of OBRSK applied on {C1,C2} that d≥α2c^>⋯>α1. On the other hand, since a<pl<b, it follows from duality that c>q2c^+1-l>d. Hence combining all these, we have c>q2c^+1-l>d≥α2c~>α1. So for z such that q2c^+1-l≤z<c, the number of elements in the topmost row  of PC1,C2 which are ≤z is 2c~.
Since{C1,C2} satisfies condition (##) with respect to the bitableau (P′,Q′) and the integer l, it follows that there exists an integer χl(≥l) such that the χlth entry (counting from left to right) of the topmost row of PC1,C2 is a. Hence it follows from duality that the backward χlth entry (i.e., the χlth entry counting from right to left) of the topmost row of QC1,C2 is c. Therefore for z such that q2c^+1-l≤z<c, the number of elements in the topmost row of QC1,C2 which are ≤z is equal to X0, where X0 is some nonnegative integer such that X0≤2c~-χl. But χl≥l, hence -χl≤-l, and therefore X0≤2c~-χl≤2c~-l.
Therefore, the number of elements in(ℕ∖{topmostrowofQC1,C2}) which are ≤z is z-X0. Hence |({topmostrowofPC1,C2}∪˙(ℕ∖{topmostrowofQC1,C2}))≤z| = 2c~+z-X0≥2c~+z-(2c~-l)=z+l=|(P1′∪˙(ℕ∖Q1′))≤z|.



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*Source: 392437-2013-03-26.xml*