Littlewood–Richardson polynomials
A. I. Molev
School of Mathematics and Statistics
University of Sydney, NSW 2006, Australia
alexm@maths.usyd.edu.au
Abstract
We introduce a family of rings of symmetric functions depending on an in-
finite sequence of parameters. A distinguished basis of such a ring is comprised
by analogues of the Schur functions. The corresponding structure coefficients
are polynomials in the parameters which we call the Littlewood–Richardson
polynomials. We give a combinatorial rule for their calculation by modifying
an earlier result of B. Sagan and the author. The new rule provides a formula
for these polynomials which is manifestly positive in the sense of W. Graham.
We apply this formula for the calculation of the product of equivariant Schu-
bert classes on Grassmannians which implies a stability property of the struc-
ture coefficients. The first manifestly positive formula for such an expansion
was given by A. Knutson and T. Tao by using combinatorics of puzzles while
the stability property was not apparent from that formula. We also use the
Littlewood–Richardson polynomials to describe the multiplication rule in the
algebra of the Casimir elements for the general linear Lie algebra in the basis
of the quantum immanants constructed by A. Okounkov and G. Olshanski.
http://arxiv.org/abs/0704.0065v3
1 Introduction
Let a = (ai), i ∈ Z be a sequence of variables. Consider the ring of polynomials
Z[a] in the variables ai with integer coefficients. Introduce another infinite set of
variables x = (x1, x2, . . . ) and for each nonnegative integer n denote by Λn the ring
of symmetric polynomials in x1, . . . , xn with coefficients in Z[a]. The ring Λn is
filtered by the usual degrees of polynomials in x1, . . . , xn with the ai considered to
have the zero degree. The evaluation map
ϕn : Λn → Λn−1, P (x1, . . . , xn) 7→ P (x1, . . . , xn−1, an) (1.1)
is a homomorphism of filtered rings so that we can define the inverse limit ring Λ by
Λ = lim
Λn, n → ∞, (1.2)
where the limit is taken with respect to the homomorphisms (1.1) in the category of
filtered rings. When a is specialized to the sequence of zeros, this reduces to the usual
definition of the ring of symmetric functions; see e.g. Macdonald [14]. In that case,
a distinguished basis of Λ is comprised by the Schur functions sλ(x) parameterized
by all partitions λ. The respective analogues of the sλ(x) in the general case are the
double Schur functions sλ(x||a) which form a basis of Λ over Z[a]. We introduce the
Littlewood–Richardson polynomials cνλµ(a) as the structure coefficients of the ring Λ
in the basis of double Schur functions,
sλ(x||a) sµ(x||a) =
cνλµ(a) sν(x||a). (1.3)
In the specialization a = (0) the polynomials cνλµ(a) become the classical Littlewood–
Richardson coefficients cνλµ; see [12]. These are remarkable nonnegative integers which
occupy a prominent place in combinatorics, representation theory and geometry; see
e.g. Fulton [5], Macdonald [14] and Sagan [21].
The main result of this paper is a combinatorial rule for the calculation of the
Littlewood–Richardson polynomials which provides a manifestly positive formula in
the sense that cνλµ(a) is written as a polynomial in the differences ai − aj , i < j, with
positive integer coefficients.
We consider two applications of the rule. The results of Knutson and Tao [9]
imply that under an appropriate specialization, the polynomials cνλµ(a) describe the
multiplication rule for the equivariant Schubert classes on Grassmannians; see also
Fulton [6] for a more direct argument. Let n and N be nonnegative integers with
n 6 N and let Gr(n,N) denote the Grassmannian of the n-dimensional vector sub-
spaces of CN . The torus T = (C∗)N acts naturally on Gr(n,N). The equivariant
cohomology ring H∗T (Gr(n,N)) is a module over the polynomial ring Z[t1, . . . , tN ]
which can be identified with H∗T ({pt}), the equivariant cohomology ring of a point.
This module has a basis of the equivariant Schubert classes σλ parameterized by all
diagrams λ contained in the n×m rectangle, m = N − n; see e.g. [5, 6]. Then
σλ σµ =
d νλµ σν , (1.4)
where d νλµ = c
λµ(a) with the sequence a specialized by
a−m+1 = −t1, a−m+2 = −t2, . . . , an = −tN , (1.5)
while the remaining parameters ai are set to zero (the ti should be replaced with
yi in the notation of [9]). The coefficients d
λµ are given explicitly as polynomials
in the ti − tj , i > j, with positive integer coefficients. This positivity property was
established by Graham [8] in the general context of the equivariant Schubert calculus.
The first manifestly positive formula for the coefficients in the expansion (1.4) was
obtained by Knutson and Tao [9] by using combinatorics of puzzles. An earlier rule
of Molev and Sagan [17] also calculates d νλµ but lacks the explicit positivity property.
Our new rule implies a stability property of the coefficients d νλµ (see Corollary 3.1
below). Even though this property was not pointed out in [9], it can be derived
directly from the puzzle rule; see also Fulton [6] for its geometrical interpretation and
an extension to the equivariant Schubert calculus on the flag variety.
As another application, we obtain a rule for the positive integer expansion of the
product of two (virtual) quantum immanants (or the corresponding higher Capelli
operators) of Okounkov and Olshanski [18, 19]; cf. [17]. The quantum immanants
Sλ|n are elements of the center Z(gln) of the universal enveloping algebra U(gln)
parameterized by partitions λ with at most n parts; see [18]. The elements Sλ|n form
a basis of Z(gln) so that we can define the coefficients f
λµ by the expansion
Sλ|n Sµ|n =
f νλµ Sν|n.
Then f νλµ = c
λµ(a) for the specialization ai = −i for i ∈ Z. As n → ∞ this yields
a multiplication rule for the virtual quantum immanants Sλ; see Section 3.2 for the
definitions.
We define the double Schur function sλ(x||a) as the sequence of the double Schur
polynomials
sλ(x1, . . . , xn ||a), n = 1, 2, . . . , (1.6)
which are compatible with respect to the homomorphisms (1.1),
ϕn : sλ(x1, . . . , xn ||a) 7→ sλ(x1, . . . , xn−1 ||a). (1.7)
The polynomials (1.6) are closely related to the “factorial” or “double” Schur poly-
nomials sλ(x|u) with x = (x1, . . . , xn). The latter were introduced by Goulden and
Greene [7] and Macdonald [13] as a generalization of the factorial Schur polynomials
of Biedenharn and Louck [1, 2], and they are also a special case of the double Schu-
bert polynomials of Lascoux and Schützenberger; see Lascoux [11]. We follow Chen,
Li and Louck [4] and Fulton [6] and use the name “double Schur polynomials” for
the related polynomials sλ(x||a) as well.
In a more detail, consider a partition λ which is a sequence λ = (λ1, . . . , λn)
of integers λi such that λ1 > · · · > λn > 0. We will identify λ with its diagram
represented graphically as the array of left justified rows of unit boxes with λ1 boxes
in the top row, λ2 boxes in the second row, etc. The total number of boxes in λ will
be denoted by |λ|. The transposed diagram λ′ = (λ′1, . . . , λ
p) is obtained from λ by
applying the symmetry with respect to the main diagonal, so that λ′j is the number
of boxes in the j-th column of λ.
Let u = (u1, u2, . . . ) be a sequence of variables. The polynomials sλ(x|u) can be
defined by
sλ(x|u) =
(xT (α) − uT (α)+c(α)), (1.8)
where T runs over all semistandard (column-strict) tableaux of shape λ with entries
in {1, . . . , n}, T (α) is the entry of T in the box α ∈ λ and c(α) = j − i is the content
of the box α = (i, j) in row i and column j.
By a reverse λ-tableau T we will mean the tableau obtained by filling in the boxes
of λ with the numbers 1, 2, . . . , n in such a way that the entries weakly decrease along
the rows and strictly decrease down the columns. If α = (i, j) is a box of λ we let
T (α) = T (i, j) denote the entry of T in the box α. We define the double Schur
polynomials sλ(x||a) by
sλ(x||a) =
(xT (α) − aT (α)−c(α)), (1.9)
summed over the reverse λ-tableaux T . Then we have
sλ(x||a) = sλ(x|u) (1.10)
for the sequences a and u related by an−i+1 = ui with i = 1, 2, . . . . In particular, the
polynomial sλ(x||a) only depends on the variables ai with i 6 n, i ∈ Z. The relation
(1.10) is verified easily by replacing xi with xn−i+1 in (1.8) for all i = 1, . . . , n and using
the fact that sλ(x|u) is a symmetric polynomial in x. The property (1.7) of the double
Schur polynomials is immediate from their definition. In the specialization of the
sequence a with ai = −i, i ∈ Z, formula (1.9) defines the shifted Schur polynomials
of Okounkov and Olshanski [18, 19] in the variables yi = xi+ i. The use of the reverse
tableaux was significant in their study of the vanishing and stability properties of
these polynomials and associated central elements of the universal enveloping algebra
for the Lie algebra gln; see also Section 3.2 below.
Note that the stability property (1.7) extends to the double Schubert polynomials
(and to the equivariant Schubert calculus on the flag manifold). This follows easily
from the Cauchy formula for the Schubert polynomials (e.g., put x1 = y1 in [15,
Formula in 2.5.5]). In a more general context, this was also pointed out in [3].
The double Schur polynomials sλ(x||a) parameterized by the diagrams λ with at
most n rows form a basis of the ring Λn. Due to the stability property (1.7), the
Littlewood–Richardson polynomials cνλµ(a) can be defined by the expansion (1.3),
where x is understood as the set of variables x = (x1, . . . , xn) for any positive integer
n such that the diagrams λ, µ and ν have at most n rows. This allows us to work
with a finite set of variables for the determination of the polynomials cνλµ(a). For
the proof of the main theorem (Theorem 2.1) we follow the general approach of [17],
using the techniques of “barred” tableaux and modify the corresponding arguments
in order to obtain manifestly positive polynomials. This is achieved by imposing a
boundness condition on the barred tableaux.
It was observed by Goulden and Greene [7] and Macdonald [13] that sλ(x|u),
regarded as a formal power series in the infinite sets of variables x and u, admits
a “supertableaux” representation. We show that this representation has its “finite”
counterpart where x is a finite set of variables. We derive the corresponding formula
by choosing a certain specialization of the 9th Variation in [13]. This representa-
tion leads to a “supertableau” expression for the Littlewood–Richardson polynomials
cνλµ(a), although that expression is neither manifestly positive, nor stable.
After the first version of this paper was completed we have learned of an indepen-
dent work of V. Kreiman [10], where a positive equivariant Littlewood–Richardson
rule was given. That rule is equivalent to our Theorem 2.1 although the proof in [10]
is different. Moreover, Kreiman’s paper also provides a weight-preserving bijection
between the Knutson–Tao puzzles and the barred tableaux used in Theorem 2.1.
This work was inspired by Bill Fulton’s lectures [6]. I am grateful to Bill for
stimulating discussions.
2 Multiplication rule
Let R denote a sequence of diagrams
µ = ρ(0) → ρ(1) → · · · → ρ(l−1) → ρ(l) = ν, (2.1)
where ρ → σ means that σ is obtained from ρ by adding one box. Let ri denote
the row number of the box added to the diagram ρ(i−1). The sequence r1r2 . . . rl is
called the Yamanouchi symbol of R. Introduce the ordering on the set of boxes of
a diagram λ by reading them by columns from left to right and from bottom to top
in each column. We call this the column order . We shall write α ≺ β if α (strictly)
precedes β with respect to the column order. Given a sequence R, construct the
set T (λ,R) of barred reverse λ-tableaux T with entries from {1, 2, . . . } such that T
contains boxes α1, . . . , αl with
α1 ≺ · · · ≺ αl and T (αi) = ri, 1 6 i 6 l.
We will distinguish the entries in α1, . . . , αl by barring each of them. So, an element
of T (λ,R) is a pair consisting of a reverse λ-tableau and a chosen sequence of barred
entries compatible with R. We shall keep the notation T for such a pair. For example,
let R be the sequence
(3, 1) → (3, 2) → (3, 2, 1) → (3, 3, 1) → (4, 3, 1)
so that the Yamanouchi symbol is 2 3 2 1. Then for λ = (5, 5, 3) the following barred
λ-tableau belongs to T (λ,R):
For each box α with αi ≺ α ≺ αi+1, 0 6 i 6 l, set ρ(α) = ρ
(i). The barred entries
r1, . . . , rl divide the tableau into regions marked by the elements of the sequence R,
as illustrated:
ρ(0) ρ(1)
r1 r2
· · ·
Finally, a reverse λ-tableau T will be called ν-bounded if
T (1, j) 6 ν ′j for all j = 1, . . . , λ1.
Note that ν-bounded λ-tableaux exist only if λ ⊆ ν.
We are now in a position to state a rule for the calculation of the Littlewood-
Richardson polynomials cνλµ(a) defined by (1.3).
Theorem 2.1. The polynomial cνλµ(a) is zero unless µ ⊆ ν. If µ ⊆ ν then
cνλµ(a) =
T (α) unbarred
aT (α)−ρ(α)
T (α)
− aT (α)−c(α)
, (2.2)
summed over all sequences R of the form (2.1) and all ν-bounded reverse λ-tableaux
T ∈ T (λ,R). Moreover, for each factor occurring in the formula (2.2) we have
ρ(α)T (α) > c(α).
Before proving the theorem, let us point out some properties of the Littlewood-
Richardson polynomials which are immediate from the rule and consider some exam-
ples. The polynomial cνλµ(a) is zero unless both diagrams λ and µ are contained in
ν and |λ| + |µ| > |ν|. In this case cνλµ(a) is a homogeneous polynomial in the ai of
degree |λ| + |µ| − |ν|. If |λ| + |µ| − |ν| = 0 then the theorem reproduces a version
of the classical Littlewood-Richardson rule; see Corollary 2.9 below. Note also that
by the definition, the polynomials have the symmetry cνλµ(a) = c
µλ(a) which is not
apparent from the rule.
Example 2.2. For the product of the double Schur functions s(2)(x||a) and s(2,1)(x||a)
we have
s(2)(x||a) s(2,1)(x||a) = s(4,1)(x||a) + s(3,2)(x||a) + s(3,1,1)(x||a) + s(2,2,1)(x||a)
a−1 − a2 + a−2 − a0
s(3,1)(x||a) +
a−1 − a2
s(2,2)(x||a)
a−1 − a0
s(2,1,1)(x||a) +
a−1 − a2
a−1 − a0
s(2,1)(x||a).
For instance, the coefficient of s(3,1)(x||a) is calculated by the following barred (2)-
tableaux
1 1 1 1 2 1
compatible with the sequence (2, 1) → (3, 1). They contribute respectively a−1 − a1,
a−2 − a0, a1 − a2 which sums up to the coefficient a−1 − a2 + a−2 − a0. Alternatively,
using the symmetry cνλµ(a) = c
µλ(a) we can calculate the coefficient of s(3,1)(x||a) by
considering the barred (2, 1)-tableaux
compatible with the sequences (2) → (3) → (3, 1) and (2) → (2, 1) → (3, 1), respec-
tively. Their contributions to the coefficient are a−2 − a0 and a−1 − a2.
Example 2.3. For the calculation of c
(5,2,2)
(4,2,1)(2,2)
(a) take λ = (4, 2, 1), µ = (2, 2) and
ν = (5, 2, 2). We have ten sequences R of the form (2.1) but the set T (λ,R) contains
ν-bounded tableaux only for three of them. For the sequence R1 with the Yamanouchi
symbol 1 3 3 1 1, the set T (λ,R1) contains two bounded barred tableaux
3 1 1
3 1 1
whose contributions to the Littlewood–Richardson polynomial are (a0 − a3)(a0 − a2)
and (a0 − a3)(a−2 − a1), respectively. For the sequence R2 with the Yamanouchi
symbol 1 3 1 3 1, the set T (λ,R2) contains the bounded tableaux
3 1 1
3 1 1
with the respective contributions (a0 − a3)(a−4 − a−2) and (a0 − a3)(a−3 − a−1). For
the sequence R3 with the Yamanouchi symbol 3 1 3 1 1, the set T (λ,R3) contains the
only bounded tableau
3 1 1
with the contribution (a−1 − a3)(a0 − a3). Hence,
(5,2,2)
(4,2,1)(2,2)
(a) = (a0 − a3) (a−4 + a−3 + a0 − a1 − a2 − a3).
Taking λ = (2, 2), µ = (4, 2, 1) and ν = (5, 2, 2) we get two sequences with the
Yamanouchi symbols 1 3 and 3 1. The corresponding sets T (λ,R) consist of five
and four bounded barred tableaux, respectively, thus leading to a slightly longer
calculation.
Proof of Theorem 2.1. We present the proof as a sequence of lemmas. Due to the
stability property (1.7), we may (and will) work with a finite set of variables x =
(x1, . . . , xn). Accordingly, possible entries of the tableaux are now elements of the set
{1, . . . , n}. Introduce another sequence of variables b = (bi), i ∈ Z, and define the
Littlewood–Richardson type coefficients cνλµ(a, b) by the expansion
sλ(x||b) sµ(x||a) =
cνλµ(a, b) sν(x||a). (2.3)
Lemma 2.4. The coefficient cνλµ(a, b) is zero unless µ ⊆ ν. If µ ⊆ ν then
cνλµ(a, b) =
T (α) unbarred
aT (α)−ρ(α)
T (α)
− bT (α)−c(α)
, (2.4)
summed over all sequences R of the form (2.1) and all reverse λ-tableaux T ∈ T (λ,R).
Proof. This is essentially a reformulation of the main result of [17] (Theorem 3.1).
Note that the summation in (2.4) is taken over all barred tableaux T ∈ T (λ,R) (not
just over the ν-bounded ones as in (2.2)). Rather than repeating the whole argument
of [17], we only sketch the main steps of the proof and indicate the necessary changes
to be made. We refer the reader to [17] for the details.
We assume that all diagrams here have at most n rows. If ρ = (ρ1, . . . , ρn) is a
such diagram, we set
aρ = (a1−ρ1 , . . . , an−ρn) and |aρ| = a1−ρ1 + · · ·+ an−ρn.
Under the correspondence (1.10) we have aρ = uρ = (uρ1+n, . . . , uρn+1), the latter
notation was used in [17].
The starting point is the Vanishing Theorem of [18] whose proof was also repro-
duced in [17]. By that theorem,
sλ(aρ ||a) = 0 unless λ ⊆ ρ,
sλ(aλ ||a) =
(i,j)∈λ
ai−λi − aλ′j−j+1
The first claim of the lemma follows from the Vanishing Theorem which also implies
λµ(a, b) = sλ(aµ ||b).
This proves (2.4) for the case ν = µ. Now we suppose that |ν| − |µ| > 1 and proceed
by induction on |ν| − |µ|. The induction step is based on the recurrence relation
cνλµ(a, b) =
|aν | − |aµ|
cνλµ+(a, b)−
λµ (a, b)
(2.5)
which was proved in [17, Proposition 3.4]; see also [9]. Suppose that the diagram ν
is obtained from µ by adding one box in row r. Then
cνλµ(a, b) =
sλ(aν ||b)− sλ(aµ ||b)
(aν)r − (aµ)r
. (2.6)
Now use the definition (1.9) of the double Schur polynomials. Since the n-tuples aν
and aµ only differ at the r-th component, the ratio on the right hand side of (2.6)
can be expanded by taking into account the entries r of the reverse λ-tableaux T .
We need the following formula, where we are thinking of y = (aν)r, z = (aµ)r and
mi = bT (α)−c(α) as α runs over the boxes of T with T (α) = r in column order:
i=1(y −mi)−
i=1(z −mi)
y − z
(z −m1) . . . (z −mj−1)(y −mj+1) . . . (y −mk).
The right hand side of (2.6) can now be interpreted as the right hand side of (2.4),
where R is the only sequence µ → ν and the sum is taken over the reverse λ-tableaux
T with one barred entry r, as illustrated:
µ r ν
Here ρ(α) = µ for all boxes α preceding the box occupied by the barred r, and
ρ(α) = ν for all boxes α which follow that box in column order. Note that the
variables y and z are now swapped on the right hand side of the above expansion,
as compared to [17] (this does not change the polynomial due to the symmetry in y
and z). Consequently, the column order used in [17] is the opposite to the order on
the boxes of λ we use here.
We can represent the above calculation of cνλµ(a) by the “diagrammatic” relation
|aν | − |aµ|
µ r ν = ν − µ
Consider now the next case where |ν| − |µ| = 2 and apply the recurrence relation
(2.5). We have three subcases: the diagram ν is obtained from µ by adding two boxes
in different rows and columns; by adding two boxes in the same row; or by adding
two boxes in the same column. The first two subcases are dealt with in a way similar
to the case |ν|− |µ| = 1. An additional care is needed for the third subcase where we
suppose that ν is obtained from µ by adding the boxes in rows r and r + 1. Denote
by ρ the diagram obtained from µ by adding the box in row r. Then (2.5) gives
cνλµ(a, b) =
cνλρ(a, b)− c
λµ(a, b)
|aν | − |aµ|
Set s = r+1. Exactly as in the case |ν|−|µ| = 1, we have the following diagrammatic
relations:
|aρ| − |aµ|
sρ ν = ρ s ν − µ s ν
|aν | − |aρ|
sρ ν = µ r ν − µ r ρ
Hence, the desired formula for cνλµ(a, b) will follow if we prove the relation
µ r ν = µ s ν
We construct a weight-preserving bijection between the barred reverse λ-tableaux
which are represented by the left and right hand sides of this diagrammatic relation.
Here the weight is the product on the right hand side of (2.4) corresponding to a
barred tableau. Let such a tableau with a barred entry r in the box (i, j) be given.
Suppose first that the box (i − 1, j) belongs to the diagram and it is occupied by
s = r+1. Then the image of the tableau under the map is the same tableau but the
entry T (i, j) = r is now unbarred while T (i− 1, j) = r + 1 is barred. Since
(aν)r+1 = (aµ)r and T (i− 1, j)− c(i− 1, j) = T (i, j)− c(i, j),
the weights of the tableaux are preserved under the map.
Suppose now that the entry in the box (i − 1, j) is greater than r + 1, or this
box is outside the diagram. Consider all entries r in the row i to the left of the box
(i, j) and suppose that they occupy the boxes (i, j −m), (i, j −m+1), . . . , (i, j− 1).
Then the image of the tableau under the map is the tableau obtained by replacing
the entries in each of the boxes (i, j −m), . . . , (i, j) with s = r + 1 and barring the
entry in the box (i, j −m). The weights of the tableaux are again preserved.
The inverse map is described in a similar way. This gives the desired weight-
preserving bijection. The general argument uses similar calculations with the barred
diagrams and a similar bijection described in [17].
Remark 2.5. (i) A cohomological interpretation of the coefficients cνλµ(a, b) and their
puzzle computation can be found in [9].
(ii) The definition (2.3) of the coefficients cνλµ(a, b) can be extended to the case
where λ is a skew diagram. Lemma 2.4 and its proof remain valid; see [17].
(iii) In contrast with the Littlewood–Richardson polynomials cνλµ(a), the coeffi-
cients cνλµ(a, b) do not have the stability property as they depend on n.
Lemma 2.4 implies that the Littlewood–Richardson polynomials can be calcu-
lated by (2.4) with b = a, that is, cνλµ(a) = c
λµ(a, a). Our strategy now is to show
that (unlike the formula of Theorem 3.1 in [17]), the formula (2.4) (with b = a) is
“nonnegative” in the sense that all nonzero products which occur in the formula are
polynomials in the ai − aj with i < j. Then we demonstrate that the ν-boundness
condition serves to eliminate the unwanted zero terms.
Lemma 2.6. Let R be a sequence of the form (2.1) and let T ∈ T (λ,R). Suppose
that ∏
T (α) unbarred
aT (α)−ρ(α)
T (α)
− aT (α)−c(α)
6= 0. (2.7)
Then ρ(α)
T (α)
> c(α) for all α ∈ λ with unbarred T (α).
Proof. Suppose on the contrary that there exists a box α = (i, j) with an unbarred
T (i, j) and the condition ρ(i, j)T (i,j) < j − i; the equality ρ(i, j)T (i,j) = j − i is
excluded since this would violate (2.7). Choose such a box with the minimum possible
value of j. If all the entries T (i, 1), . . . , T (i, j − 1) of T are barred then ρ(i, j) is
obtained from µ by adding boxes in rows T (i, 1) > · · · > T (i, j − 1) and, possibly,
by adding other boxes. Since T (i, j − 1) > T (i, j), we have ρ(i, j)T (i,j) > j − 1, a
contradiction. So, at least one of the entries T (i, 1), . . . , T (i, j−1) must be unbarred.
Take such an unbarred entry T (i, k) which is the closest to T (i, j), that is, all entries
T (i, k+1), . . . , T (i, j − 1) are barred. Then ρ(i, j) is obtained from ρ(i, k) by adding
boxes in rows T (i, k + 1) > · · · > T (i, j − 1) and, possibly, by adding other boxes.
Hence,
ρ(i, j)T (i,j) > ρ(i, k)T (i,k) + j − k − 1
which implies ρ(i, k)T (i,k) < k − i + 1. However, if ρ(i, k)T (i,k) = k − i then the
factor in (2.7) corresponding to α = (i, k) is zero, which is impossible. Therefore
ρ(i, k)T (i,k) < k − i which contradicts the choice of j.
Lemma 2.7. Suppose that R is a sequence of the form (2.1) and T ∈ T (λ,R). If
(2.7) holds then T is ν-bounded.
Proof. By Lemma 2.6, for all unbarred entries T (1, k) of the first row of the tableau
T we have ρ(1, k)T (1,k) > k. This implies νT (1,k) > k. If the entry T (1, j) is barred
then ρ(1, k)T (1,k) > k for the nearest unbarred entry T (1, k) on its left (if it exists).
Then ν is obtained from ρ(1, k) by adding boxes in rows T (1, k + 1) > · · · > T (1, j)
and, possibly, by adding other boxes. This implies νT (1,j) > j. Thus, this inequality
holds for all j = 1, . . . , λ1. This is equivalent to the ν-boundness of T .
Lemma 2.8. Suppose that R is a sequence of the form (2.1) and T ∈ T (λ,R) is
ν-bounded. Then ρ(α)
T (α)
> c(α) for all α ∈ λ with unbarred T (α).
Proof. We argue by contradiction. Taking into account Lemma 2.6, we find that for
some α = (i, j) with unbarred T (α) we have ρ(i, j)T (i,j) = j − i. Set t = T (i, j)
and consider all barred entries of T (assuming for now they exist) which are equal
to t and occur to the right of the column j. Since T is a reverse tableau, these
entries t̄ can only occur in rows 1, 2, . . . , i. Let (r, k) be the box with the maximum
column number k containing t̄. Then the total number of such entries t̄ does not
exceed k− j. This implies that the number of boxes νt in row t of ν does not exceed
ρ(i, j)t + k − j = k − i. Hence, ν
k 6 t − 1. On the other hand, by the ν-boundness
of T we have t = T (r, k) 6 T (1, k) 6 ν ′k, a contradiction.
If none of the boxes to the right of the column j contains t̄ then νt = ρ(i, j)t = j−i.
However, by the assumption, νt > νT (1,j) > j, a contradiction.
This completes the proof of the theorem.
By the column word of a tableau T we will mean the sequence of all entries of T
written in the column order.
Corollary 2.9. Suppose that |ν| = |λ| + |µ|. The Littlewood–Richardson coefficient
cνλµ equals the number of ν-bounded reverse λ-tableaux T whose column word coincides
with the Yamanouchi symbol of a certain sequence R of the form (2.1).
This can be shown to be equivalent to a well-known version of the Littlewood–
Richardson rule. Corollary 2.9 also holds with the ν-boundness condition dropped;
see Lemma 2.7. By the corollary, cνλµ counts the cardinality of the intersection of two
finite sets: the set of column words of ν-bounded reverse λ-tableaux and the set of
Yamanouchi symbols of the sequences of the form (2.1).
Remark 2.10. Due to (1.10), the multiplication rule for the polynomials sλ(x|u) is
obtained from Theorem 2.1 by replacing ai with un−i+1 for each i. The corresponding
coefficients are polynomials in the ui−uj, i > j, with positive integer coefficients.
Corollary 2.11. Suppose that the polynomials cνλµ(a) are defined by the expansion
(1.3) with x = (x1, . . . , xn). Then c
λµ(a) is independent of n as soon as n > ν
Moreover, if n < ν ′1 then c
λµ(a) = 0.
Proof. This follows from the boundness condition on the reverse tableaux.
3 Applications
3.1 Equivariant Schubert calculus on the Grassmannian
As in the Introduction, consider the equivariant cohomology ring H∗T (Gr(n,N)) as
a module over Z[t1, . . . , tN ]. Let x1, . . . , xn denote the Chern roots of the dual S
of the tautological subbundle S of the trivial bundle CNGr(n,N) so that for the total
equivariant Chern class of S we have
cT (S) =
(1− xi).
Then, due to [6, Lecture 8, Proposition 1.1] (see also [16]), the equivariant Schubert
classes σλ can be expressed by
σλ = sλ(x|u), u = (−tN , . . . ,−t1, 0, . . . ).
Hence, Theorem 2.1 yields a multiplication rule for the equivariant Schubert classes.
The corresponding stability property is implied by Corollary 2.11.
Corollary 3.1. We have
σλ σµ =
d νλµ σν ,
where
d νλµ =
T (α) unbarred
tm+T (α)−c(α) − tm+T (α)−ρ(α)
T (α)
, (3.1)
summed over all sequences R of the form (2.1) and all ν-bounded reverse λ-tableaux
T ∈ T (λ,R). In particular, the d νλµ are polynomials in the ti− tj, i > j, with positive
integer coefficients. Moreover, the coefficients d νλµ, regarded as polynomials in the
variables ai defined in (1.5), are independent of n and m, as soon as the inequalities
n > λ′1 + µ
1 and m > λ1 + µ1 hold.
Example 3.2. For any n > 3 and m > 4 we have
σ(2) σ(2,1) = σ(4,1) + σ(3,2) + σ(3,1,1) + σ(2,2,1)
+ (tm+2 − tm−1 + tm − tm−2) σ(3,1) + (tm+2 − tm−1) σ(2,2)
+ (tm − tm−1) σ(2,1,1) + (tm+2 − tm−1) (tm − tm−1) σ(2,1).
This follows from Example 2.2.
The first manifestly positive rule for the expansion of σλ σµ was given by Knutson
and Tao [9] by using combinatorics of puzzles. Although the stability property was not
pointed out in [9], it can be deduced directly from the puzzle rule or by applying the
weight-preserving bijection between the puzzles and the barred tableaux constructed
by Kreiman [10].
3.2 Quantum immanants and higher Capelli operators
Let gln denote the general linear Lie algebra over C. Consider the center Z(gln) of
the universal enveloping algebra U(gln). The algebra U(gln) is equipped with the
natural filtration. For all n we identify gln−1 as a subalgebra of gln in a usual way
and denote by gl∞ the corresponding inductive limit
gl∞ =
Due to Olshanski [20], there exist filtration-preserving homomorphisms
on : Z(gln) → Z(gln−1), n > 1, (3.2)
which allow one to define the algebra Z of the virtual Casimir elements for the Lie
algebra gl∞ as the inverse limit
Z = lim
Z(gln), n → ∞,
in the category of filtered algebras.
The quantum immanants Sλ|n are elements of the center Z(gln) of the universal
enveloping algebra U(gln) parameterized by the diagrams λ with at most n rows;
see [18]. The elements Sλ|n form a basis of Z(gln) and they are consistent with the
Olshanski homomorphisms (3.2) so that
on : Sλ|n 7→ Sλ|n−1, (3.3)
where we assume Sλ|n = 0 if the number of rows of λ exceeds n. For any diagram λ,
the corresponding virtual quantum immanant Sλ is then defined as the sequence
Sλ = ( Sλ|n |n > 0).
The elements Sλ parameterized by all diagrams λ form a basis of the algebra Z so
that we can define the coefficients f νλµ by the expansion
Sλ Sµ =
f νλµ Sν .
Note that the same coefficients f νλµ determine the multiplication rule for the higher
Capelli operators ∆λ, which are defined as the sequences of the images of the quantum
immanants Sλ|n, where each image is taken under a natural representation of gln by
differential operators; see [18, 19].
Corollary 3.3. The coefficient f νλµ is zero unless µ ⊆ ν. If µ ⊆ ν then
f νλµ =
T (α) unbarred
ρ(α)T (α) − c(α)
, (3.4)
summed over all sequences R of the form (2.1) and all ν-bounded reverse λ-tableaux
T ∈ T (λ,R). In particular, the f νλµ are nonnegative integers.
Proof. Due to the stability property (3.3) of the quantum immanants, it suffices to
calculate the corresponding coefficients for the expansion of the products Sλ|n Sµ|n.
The images of the quantum immanants Sλ|n under the Harish-Chandra isomorphism
can be identified with the double Schur polynomials sλ(x||a) where the sequence a is
specialized to ai = −i; see [18]. Therefore, the coefficients in question coincide with
the corresponding specializations of the Littlewood–Richardson polynomials cνλµ(a).
Example 3.4. Using Example 2.2 we get
S(2) S(2,1) = S(4,1) + S(3,2) + S(3,1,1) + S(2,2,1) + 5 S(3,1) + 3 S(2,2) + S(2,1,1) + 3 S(2,1).
In the course of the proof of Corollary 3.3 we also calculated the coefficients
for the expansion of the products Sλ|n Sµ|n for any n. Some other formulas for these
coefficients were obtained in [17]. In particular, it was shown that the f νλµ are integers,
although their positivity property was not established there.
Note also that the algebra of virtual Casimir elements Z is isomorphic to the
algebra of shifted symmetric functions Λ∗; see [19]. The latter can be regarded as the
specialization of Λ (or rather, its extension over C) at ai = −i for all i ∈ Z.
4 Supertableau formulas for sλ(x||a) and c
λµ(a)
Here we obtain one more rule for the calculation of the Littlewood–Richardson poly-
nomials cνλµ(a). It relies on a supertableau representation of the double Schur poly-
nomials sλ(x||a) which is implied by the results of [13]. This representation provides
a “finite” version of the supertableau formulas of [7] and [13]; cf. [4].
Fix a positive integer n. For r > 1 set u(r) = (u1, . . . , ur) and use the 9th Variation
in [13] with the indeterminates hrs specialized by
hrs = hr(u
(n−r−s+1)) if r + s 6 n,
and 0 otherwise, where hr denotes the r-th complete symmetric polynomial. Let us
write ŝλ/µ(u) for the corresponding Schur functions. Then (8.2) and (9.1) in [13] give
ŝλ/µ(u) =
α∈λ/µ
uT (α),
summed over semistandard tableaux T of shape λ/µ, such that the entries of the i-th
row do not exceed n− λi + i. Furthermore, using (6.18)
1 and (9.6 ′) in [13] we get
sλ(x|u) =
sµ(x) ŝλ′/µ′(−u). (4.5)
Equivalently, this can be interpreted as a combinatorial expression for the polynomials
sλ(x|u) in terms of “supertableaux”. Identify the indices of u with the symbols
1′, 2′, . . . . A supertableau T is obtained by filling in the diagram of λ with the indices
1, . . . , n, 1′, 2′, . . . in such a way that in each row (resp. column) each primed index is
to the right (resp. below) of each unprimed index; unprimed indices weakly increase
along the rows and strictly increase down the columns; primed indices strictly increase
along the rows and weakly increase down the columns; primed indices in column j
do not exceed n− λ′j + j. Relation (4.5) implies the following.
Proposition 4.1. We have
sλ(x|u) =
T (α) unprimed
xT (α)
T (α) primed
(−uT (α)), (4.6)
summed over all λ-supertableaux T .
Using (1.10), we get an analogous representation for the double Schur polynomials
sλ(x||a). A reverse supertableau T is obtained by filling in the diagram of λ with
1This formula in [13] should be corrected by replacing a(λj+n−j) with a(λi+n−i).
the indices 1, . . . , n, n′, (n − 1)′, . . . (including non-positive primed indices) in such
a way that in each row (resp. column) each primed index is to the right (resp.
below) of each unprimed index; unprimed indices weakly decrease along the rows and
strictly decrease down the columns; primed indices strictly decrease along the rows
and weakly decrease down the columns; primed indices in column j are not less than
λ′j − j + 1. The following supertableau representation of the polynomials sλ(x||a)
follows from Proposition 4.1.
Corollary 4.2. We have
sλ(x||a) =
T (α) unprimed
xT (α)
T (α) primed
(−aT (α)), (4.7)
summed over all reverse λ-supertableaux T .
Example 4.3. Let n = 2 and λ = (2, 1). By the definition (1.9),
s(2,1)(x||a) = (x2 − a2)(x1 − a0)(x1 − a2) + (x2 − a2)(x2 − a1)(x1 − a2).
On the other hand, the reverse (2, 1)-supertableaux are
2 0 ′
2 1 ′
2 2 ′
2 0 ′
2 1 ′
2 2 ′
1 0 ′
1 1 ′
1 2 ′
2 ′ 0 ′
2 ′ 1 ′
which yield
s(2,1)(x||a) = x
1x2 + x1x
2 − x1x2a2 − x
2a2 − x
1a2 − x1x2a0 − x1x2a1 − x1x2a2
+ x2a0a2 + x2a1a2 + x2a
2 + x1a0a2 + x1a1a2 + x1a
2 − a0a
2 − a1a
Formula (4.5) implies a supertableau representation of the coefficients cνλµ(a, b)
and hence, of the Littlewood–Richardson polynomials cνλµ(a). The representation
for the latter is neither manifestly positive, nor stable; it provides an expression for
cνλµ(a) as an alternating sum of monomials in the ai. Given a sequence R of the form
(2.1), construct the set S(λ,R) of barred reverse λ-supertableaux by analogy with
T (λ,R). A tableau T ∈ S(λ,R) must contain boxes α1, . . . , αl occupied by unprimed
indices r1, r2, . . . , rl listed in the column order which is restricted to the subtableau of
T formed by the unprimed indices. As before, we distinguish the entries in α1, . . . , αl
by barring each of them. For each box α with αi ≺ α ≺ αi+1, 0 6 i 6 l, which is
occupied by an unprimed index, set ρ(α) = ρ(i).
Corollary 4.4. The coefficients cνλµ(a, b) defined in (2.3) can be given by
cνλµ(a, b) =
T (α) unprimed, unbarred
aT (α)−ρ(α)
T (α)
T (α) primed
(−bT (α)), (4.8)
summed over sequences R of the form (2.1) and reverse supertableaux T ∈ S(λ,R).
Proof. Applying formula (4.5) we can reduce the calculation of cνλµ(a, b) to the par-
ticular case of the sequence b = (0). Now (4.8) follows from Lemma 2.4.
Example 4.5. In order to calculate the Littlewood–Richardson polynomial c
(2,1)
(2,1) (2)
we may take n = 2; see Corollary 2.11. The barred reverse supertableaux compatible
with the sequence (2) → (2, 1) are
2 0 ′
2 1 ′
2 2 ′
2 0 ′
2 1 ′
2 2 ′
so that
(2,1)
(2,1) (2)
(a) = a2−1 + a−1a1 + a−1a2 − a−1a2 − a1a2 − a
− a−1a0 − a−1a1 − a−1a2 + a0a2 + a1a2 + a
= a2−1 − a−1a0 − a−1a2 + a0a2,
which agrees with Example 2.2.
References
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in terms of tableaux, Advances in Appl. Math. 10 (1989), 396–438.
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nomials defined by tableaux, Proc. Nat. Acad. Sci. U.S.A. 87 (1990), 1441–1445.
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ans, Duke Math. J. 119 (2003), 221–260.
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SMF/AMS Texts and Monographs, Vol. 6, 1998.
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functions, Trans. Amer. Math. Soc, 351 (1999), 4429–4443.
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Groups 1 (1996), 99–126.
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Verlag, New York, 2001.
	Introduction
	Multiplication rule
	Applications
	Equivariant Schubert calculus on the Grassmannian
	Quantum immanants and higher Capelli operators
	Supertableau formulas for s(x || a) and c(a)

  We introduce a family of rings of symmetric functions depending on an
infinite sequence of parameters. A distinguished basis of such a ring is
comprised by analogues of the Schur functions. The corresponding structure
coefficients are polynomials in the parameters which we call the
Littlewood-Richardson polynomials. We give a combinatorial rule for their
calculation by modifying an earlier result of B. Sagan and the author. The new
rule provides a formula for these polynomials which is manifestly positive in
the sense of W. Graham. We apply this formula for the calculation of the
product of equivariant Schubert classes on Grassmannians which implies a
stability property of the structure coefficients. The first manifestly positive
formula for such an expansion was given by A. Knutson and T. Tao by using
combinatorics of puzzles while the stability property was not apparent from
that formula. We also use the Littlewood-Richardson polynomials to describe the
multiplication rule in the algebra of the Casimir elements for the general
linear Lie algebra in the basis of the quantum immanants constructed by A.
Okounkov and G. Olshanski.

Introduction
Let a = (ai), i ∈ Z be a sequence of variables. Consider the ring of polynomials
Z[a] in the variables ai with integer coefficients. Introduce another infinite set of
variables x = (x1, x2, . . . ) and for each nonnegative integer n denote by Λn the ring
of symmetric polynomials in x1, . . . , xn with coefficients in Z[a]. The ring Λn is
filtered by the usual degrees of polynomials in x1, . . . , xn with the ai considered to
have the zero degree. The evaluation map
ϕn : Λn → Λn−1, P (x1, . . . , xn) 7→ P (x1, . . . , xn−1, an) (1.1)
is a homomorphism of filtered rings so that we can define the inverse limit ring Λ by
Λ = lim
Λn, n → ∞, (1.2)
where the limit is taken with respect to the homomorphisms (1.1) in the category of
filtered rings. When a is specialized to the sequence of zeros, this reduces to the usual
definition of the ring of symmetric functions; see e.g. Macdonald [14]. In that case,
a distinguished basis of Λ is comprised by the Schur functions sλ(x) parameterized
by all partitions λ. The respective analogues of the sλ(x) in the general case are the
double Schur functions sλ(x||a) which form a basis of Λ over Z[a]. We introduce the
Littlewood–Richardson polynomials cνλµ(a) as the structure coefficients of the ring Λ
in the basis of double Schur functions,
sλ(x||a) sµ(x||a) =
cνλµ(a) sν(x||a). (1.3)
In the specialization a = (0) the polynomials cνλµ(a) become the classical Littlewood–
Richardson coefficients cνλµ; see [12]. These are remarkable nonnegative integers which
occupy a prominent place in combinatorics, representation theory and geometry; see
e.g. Fulton [5], Macdonald [14] and Sagan [21].
The main result of this paper is a combinatorial rule for the calculation of the
Littlewood–Richardson polynomials which provides a manifestly positive formula in
the sense that cνλµ(a) is written as a polynomial in the differences ai − aj , i < j, with
positive integer coefficients.
We consider two applications of the rule. The results of Knutson and Tao [9]
imply that under an appropriate specialization, the polynomials cνλµ(a) describe the
multiplication rule for the equivariant Schubert classes on Grassmannians; see also
Fulton [6] for a more direct argument. Let n and N be nonnegative integers with
n 6 N and let Gr(n,N) denote the Grassmannian of the n-dimensional vector sub-
spaces of CN . The torus T = (C∗)N acts naturally on Gr(n,N). The equivariant
cohomology ring H∗T (Gr(n,N)) is a module over the polynomial ring Z[t1, . . . , tN ]
which can be identified with H∗T ({pt}), the equivariant cohomology ring of a point.
This module has a basis of the equivariant Schubert classes σλ parameterized by all
diagrams λ contained in the n×m rectangle, m = N − n; see e.g. [5, 6]. Then
σλ σµ =
d νλµ σν , (1.4)
where d νλµ = c
λµ(a) with the sequence a specialized by
a−m+1 = −t1, a−m+2 = −t2, . . . , an = −tN , (1.5)
while the remaining parameters ai are set to zero (the ti should be replaced with
yi in the notation of [9]). The coefficients d
λµ are given explicitly as polynomials
in the ti − tj , i > j, with positive integer coefficients. This positivity property was
established by Graham [8] in the general context of the equivariant Schubert calculus.
The first manifestly positive formula for the coefficients in the expansion (1.4) was
obtained by Knutson and Tao [9] by using combinatorics of puzzles. An earlier rule
of Molev and Sagan [17] also calculates d νλµ but lacks the explicit positivity property.
Our new rule implies a stability property of the coefficients d νλµ (see Corollary 3.1
below). Even though this property was not pointed out in [9], it can be derived
directly from the puzzle rule; see also Fulton [6] for its geometrical interpretation and
an extension to the equivariant Schubert calculus on the flag variety.
As another application, we obtain a rule for the positive integer expansion of the
product of two (virtual) quantum immanants (or the corresponding higher Capelli
operators) of Okounkov and Olshanski [18, 19]; cf. [17]. The quantum immanants
Sλ|n are elements of the center Z(gln) of the universal enveloping algebra U(gln)
parameterized by partitions λ with at most n parts; see [18]. The elements Sλ|n form
a basis of Z(gln) so that we can define the coefficients f
λµ by the expansion
Sλ|n Sµ|n =
f νλµ Sν|n.
Then f νλµ = c
λµ(a) for the specialization ai = −i for i ∈ Z. As n → ∞ this yields
a multiplication rule for the virtual quantum immanants Sλ; see Section 3.2 for the
definitions.
We define the double Schur function sλ(x||a) as the sequence of the double Schur
polynomials
sλ(x1, . . . , xn ||a), n = 1, 2, . . . , (1.6)
which are compatible with respect to the homomorphisms (1.1),
ϕn : sλ(x1, . . . , xn ||a) 7→ sλ(x1, . . . , xn−1 ||a). (1.7)
The polynomials (1.6) are closely related to the “factorial” or “double” Schur poly-
nomials sλ(x|u) with x = (x1, . . . , xn). The latter were introduced by Goulden and
Greene [7] and Macdonald [13] as a generalization of the factorial Schur polynomials
of Biedenharn and Louck [1, 2], and they are also a special case of the double Schu-
bert polynomials of Lascoux and Schützenberger; see Lascoux [11]. We follow Chen,
Li and Louck [4] and Fulton [6] and use the name “double Schur polynomials” for
the related polynomials sλ(x||a) as well.
In a more detail, consider a partition λ which is a sequence λ = (λ1, . . . , λn)
of integers λi such that λ1 > · · · > λn > 0. We will identify λ with its diagram
represented graphically as the array of left justified rows of unit boxes with λ1 boxes
in the top row, λ2 boxes in the second row, etc. The total number of boxes in λ will
be denoted by |λ|. The transposed diagram λ′ = (λ′1, . . . , λ
p) is obtained from λ by
applying the symmetry with respect to the main diagonal, so that λ′j is the number
of boxes in the j-th column of λ.
Let u = (u1, u2, . . . ) be a sequence of variables. The polynomials sλ(x|u) can be
defined by
sλ(x|u) =
(xT (α) − uT (α)+c(α)), (1.8)
where T runs over all semistandard (column-strict) tableaux of shape λ with entries
in {1, . . . , n}, T (α) is the entry of T in the box α ∈ λ and c(α) = j − i is the content
of the box α = (i, j) in row i and column j.
By a reverse λ-tableau T we will mean the tableau obtained by filling in the boxes
of λ with the numbers 1, 2, . . . , n in such a way that the entries weakly decrease along
the rows and strictly decrease down the columns. If α = (i, j) is a box of λ we let
T (α) = T (i, j) denote the entry of T in the box α. We define the double Schur
polynomials sλ(x||a) by
sλ(x||a) =
(xT (α) − aT (α)−c(α)), (1.9)
summed over the reverse λ-tableaux T . Then we have
sλ(x||a) = sλ(x|u) (1.10)
for the sequences a and u related by an−i+1 = ui with i = 1, 2, . . . . In particular, the
polynomial sλ(x||a) only depends on the variables ai with i 6 n, i ∈ Z. The relation
(1.10) is verified easily by replacing xi with xn−i+1 in (1.8) for all i = 1, . . . , n and using
the fact that sλ(x|u) is a symmetric polynomial in x. The property (1.7) of the double
Schur polynomials is immediate from their definition. In the specialization of the
sequence a with ai = −i, i ∈ Z, formula (1.9) defines the shifted Schur polynomials
of Okounkov and Olshanski [18, 19] in the variables yi = xi+ i. The use of the reverse
tableaux was significant in their study of the vanishing and stability properties of
these polynomials and associated central elements of the universal enveloping algebra
for the Lie algebra gln; see also Section 3.2 below.
Note that the stability property (1.7) extends to the double Schubert polynomials
(and to the equivariant Schubert calculus on the flag manifold). This follows easily
from the Cauchy formula for the Schubert polynomials (e.g., put x1 = y1 in [15,
Formula in 2.5.5]). In a more general context, this was also pointed out in [3].
The double Schur polynomials sλ(x||a) parameterized by the diagrams λ with at
most n rows form a basis of the ring Λn. Due to the stability property (1.7), the
Littlewood–Richardson polynomials cνλµ(a) can be defined by the expansion (1.3),
where x is understood as the set of variables x = (x1, . . . , xn) for any positive integer
n such that the diagrams λ, µ and ν have at most n rows. This allows us to work
with a finite set of variables for the determination of the polynomials cνλµ(a). For
the proof of the main theorem (Theorem 2.1) we follow the general approach of [17],
using the techniques of “barred” tableaux and modify the corresponding arguments
in order to obtain manifestly positive polynomials. This is achieved by imposing a
boundness condition on the barred tableaux.
It was observed by Goulden and Greene [7] and Macdonald [13] that sλ(x|u),
regarded as a formal power series in the infinite sets of variables x and u, admits
a “supertableaux” representation. We show that this representation has its “finite”
counterpart where x is a finite set of variables. We derive the corresponding formula
by choosing a certain specialization of the 9th Variation in [13]. This representa-
tion leads to a “supertableau” expression for the Littlewood–Richardson polynomials
cνλµ(a), although that expression is neither manifestly positive, nor stable.
After the first version of this paper was completed we have learned of an indepen-
dent work of V. Kreiman [10], where a positive equivariant Littlewood–Richardson
rule was given. That rule is equivalent to our Theorem 2.1 although the proof in [10]
is different. Moreover, Kreiman’s paper also provides a weight-preserving bijection
between the Knutson–Tao puzzles and the barred tableaux used in Theorem 2.1.
This work was inspired by Bill Fulton’s lectures [6]. I am grateful to Bill for
stimulating discussions.
2 Multiplication rule
Let R denote a sequence of diagrams
µ = ρ(0) → ρ(1) → · · · → ρ(l−1) → ρ(l) = ν, (2.1)
where ρ → σ means that σ is obtained from ρ by adding one box. Let ri denote
the row number of the box added to the diagram ρ(i−1). The sequence r1r2 . . . rl is
called the Yamanouchi symbol of R. Introduce the ordering on the set of boxes of
a diagram λ by reading them by columns from left to right and from bottom to top
in each column. We call this the column order . We shall write α ≺ β if α (strictly)
precedes β with respect to the column order. Given a sequence R, construct the
set T (λ,R) of barred reverse λ-tableaux T with entries from {1, 2, . . . } such that T
contains boxes α1, . . . , αl with
α1 ≺ · · · ≺ αl and T (αi) = ri, 1 6 i 6 l.
We will distinguish the entries in α1, . . . , αl by barring each of them. So, an element
of T (λ,R) is a pair consisting of a reverse λ-tableau and a chosen sequence of barred
entries compatible with R. We shall keep the notation T for such a pair. For example,
let R be the sequence
(3, 1) → (3, 2) → (3, 2, 1) → (3, 3, 1) → (4, 3, 1)
so that the Yamanouchi symbol is 2 3 2 1. Then for λ = (5, 5, 3) the following barred
λ-tableau belongs to T (λ,R):
For each box α with αi ≺ α ≺ αi+1, 0 6 i 6 l, set ρ(α) = ρ
(i). The barred entries
r1, . . . , rl divide the tableau into regions marked by the elements of the sequence R,
as illustrated:
ρ(0) ρ(1)
r1 r2
· · ·
Finally, a reverse λ-tableau T will be called ν-bounded if
T (1, j) 6 ν ′j for all j = 1, . . . , λ1.
Note that ν-bounded λ-tableaux exist only if λ ⊆ ν.
We are now in a position to state a rule for the calculation of the Littlewood-
Richardson polynomials cνλµ(a) defined by (1.3).
Theorem 2.1. The polynomial cνλµ(a) is zero unless µ ⊆ ν. If µ ⊆ ν then
cνλµ(a) =
T (α) unbarred
aT (α)−ρ(α)
T (α)
− aT (α)−c(α)
, (2.2)
summed over all sequences R of the form (2.1) and all ν-bounded reverse λ-tableaux
T ∈ T (λ,R). Moreover, for each factor occurring in the formula (2.2) we have
ρ(α)T (α) > c(α).
Before proving the theorem, let us point out some properties of the Littlewood-
Richardson polynomials which are immediate from the rule and consider some exam-
ples. The polynomial cνλµ(a) is zero unless both diagrams λ and µ are contained in
ν and |λ| + |µ| > |ν|. In this case cνλµ(a) is a homogeneous polynomial in the ai of
degree |λ| + |µ| − |ν|. If |λ| + |µ| − |ν| = 0 then the theorem reproduces a version
of the classical Littlewood-Richardson rule; see Corollary 2.9 below. Note also that
by the definition, the polynomials have the symmetry cνλµ(a) = c
µλ(a) which is not
apparent from the rule.
Example 2.2. For the product of the double Schur functions s(2)(x||a) and s(2,1)(x||a)
we have
s(2)(x||a) s(2,1)(x||a) = s(4,1)(x||a) + s(3,2)(x||a) + s(3,1,1)(x||a) + s(2,2,1)(x||a)
a−1 − a2 + a−2 − a0
s(3,1)(x||a) +
a−1 − a2
s(2,2)(x||a)
a−1 − a0
s(2,1,1)(x||a) +
a−1 − a2
a−1 − a0
s(2,1)(x||a).
For instance, the coefficient of s(3,1)(x||a) is calculated by the following barred (2)-
tableaux
1 1 1 1 2 1
compatible with the sequence (2, 1) → (3, 1). They contribute respectively a−1 − a1,
a−2 − a0, a1 − a2 which sums up to the coefficient a−1 − a2 + a−2 − a0. Alternatively,
using the symmetry cνλµ(a) = c
µλ(a) we can calculate the coefficient of s(3,1)(x||a) by
considering the barred (2, 1)-tableaux
compatible with the sequences (2) → (3) → (3, 1) and (2) → (2, 1) → (3, 1), respec-
tively. Their contributions to the coefficient are a−2 − a0 and a−1 − a2.
Example 2.3. For the calculation of c
(5,2,2)
(4,2,1)(2,2)
(a) take λ = (4, 2, 1), µ = (2, 2) and
ν = (5, 2, 2). We have ten sequences R of the form (2.1) but the set T (λ,R) contains
ν-bounded tableaux only for three of them. For the sequence R1 with the Yamanouchi
symbol 1 3 3 1 1, the set T (λ,R1) contains two bounded barred tableaux
3 1 1
3 1 1
whose contributions to the Littlewood–Richardson polynomial are (a0 − a3)(a0 − a2)
and (a0 − a3)(a−2 − a1), respectively. For the sequence R2 with the Yamanouchi
symbol 1 3 1 3 1, the set T (λ,R2) contains the bounded tableaux
3 1 1
3 1 1
with the respective contributions (a0 − a3)(a−4 − a−2) and (a0 − a3)(a−3 − a−1). For
the sequence R3 with the Yamanouchi symbol 3 1 3 1 1, the set T (λ,R3) contains the
only bounded tableau
3 1 1
with the contribution (a−1 − a3)(a0 − a3). Hence,
(5,2,2)
(4,2,1)(2,2)
(a) = (a0 − a3) (a−4 + a−3 + a0 − a1 − a2 − a3).
Taking λ = (2, 2), µ = (4, 2, 1) and ν = (5, 2, 2) we get two sequences with the
Yamanouchi symbols 1 3 and 3 1. The corresponding sets T (λ,R) consist of five
and four bounded barred tableaux, respectively, thus leading to a slightly longer
calculation.
Proof of Theorem 2.1. We present the proof as a sequence of lemmas. Due to the
stability property (1.7), we may (and will) work with a finite set of variables x =
(x1, . . . , xn). Accordingly, possible entries of the tableaux are now elements of the set
{1, . . . , n}. Introduce another sequence of variables b = (bi), i ∈ Z, and define the
Littlewood–Richardson type coefficients cνλµ(a, b) by the expansion
sλ(x||b) sµ(x||a) =
cνλµ(a, b) sν(x||a). (2.3)
Lemma 2.4. The coefficient cνλµ(a, b) is zero unless µ ⊆ ν. If µ ⊆ ν then
cνλµ(a, b) =
T (α) unbarred
aT (α)−ρ(α)
T (α)
− bT (α)−c(α)
, (2.4)
summed over all sequences R of the form (2.1) and all reverse λ-tableaux T ∈ T (λ,R).
Proof. This is essentially a reformulation of the main result of [17] (Theorem 3.1).
Note that the summation in (2.4) is taken over all barred tableaux T ∈ T (λ,R) (not
just over the ν-bounded ones as in (2.2)). Rather than repeating the whole argument
of [17], we only sketch the main steps of the proof and indicate the necessary changes
to be made. We refer the reader to [17] for the details.
We assume that all diagrams here have at most n rows. If ρ = (ρ1, . . . , ρn) is a
such diagram, we set
aρ = (a1−ρ1 , . . . , an−ρn) and |aρ| = a1−ρ1 + · · ·+ an−ρn.
Under the correspondence (1.10) we have aρ = uρ = (uρ1+n, . . . , uρn+1), the latter
notation was used in [17].
The starting point is the Vanishing Theorem of [18] whose proof was also repro-
duced in [17]. By that theorem,
sλ(aρ ||a) = 0 unless λ ⊆ ρ,
sλ(aλ ||a) =
(i,j)∈λ
ai−λi − aλ′j−j+1
The first claim of the lemma follows from the Vanishing Theorem which also implies
λµ(a, b) = sλ(aµ ||b).
This proves (2.4) for the case ν = µ. Now we suppose that |ν| − |µ| > 1 and proceed
by induction on |ν| − |µ|. The induction step is based on the recurrence relation
cνλµ(a, b) =
|aν | − |aµ|
cνλµ+(a, b)−
λµ (a, b)
(2.5)
which was proved in [17, Proposition 3.4]; see also [9]. Suppose that the diagram ν
is obtained from µ by adding one box in row r. Then
cνλµ(a, b) =
sλ(aν ||b)− sλ(aµ ||b)
(aν)r − (aµ)r
. (2.6)
Now use the definition (1.9) of the double Schur polynomials. Since the n-tuples aν
and aµ only differ at the r-th component, the ratio on the right hand side of (2.6)
can be expanded by taking into account the entries r of the reverse λ-tableaux T .
We need the following formula, where we are thinking of y = (aν)r, z = (aµ)r and
mi = bT (α)−c(α) as α runs over the boxes of T with T (α) = r in column order:
i=1(y −mi)−
i=1(z −mi)
y − z
(z −m1) . . . (z −mj−1)(y −mj+1) . . . (y −mk).
The right hand side of (2.6) can now be interpreted as the right hand side of (2.4),
where R is the only sequence µ → ν and the sum is taken over the reverse λ-tableaux
T with one barred entry r, as illustrated:
µ r ν
Here ρ(α) = µ for all boxes α preceding the box occupied by the barred r, and
ρ(α) = ν for all boxes α which follow that box in column order. Note that the
variables y and z are now swapped on the right hand side of the above expansion,
as compared to [17] (this does not change the polynomial due to the symmetry in y
and z). Consequently, the column order used in [17] is the opposite to the order on
the boxes of λ we use here.
We can represent the above calculation of cνλµ(a) by the “diagrammatic” relation
|aν | − |aµ|
µ r ν = ν − µ
Consider now the next case where |ν| − |µ| = 2 and apply the recurrence relation
(2.5). We have three subcases: the diagram ν is obtained from µ by adding two boxes
in different rows and columns; by adding two boxes in the same row; or by adding
two boxes in the same column. The first two subcases are dealt with in a way similar
to the case |ν|− |µ| = 1. An additional care is needed for the third subcase where we
suppose that ν is obtained from µ by adding the boxes in rows r and r + 1. Denote
by ρ the diagram obtained from µ by adding the box in row r. Then (2.5) gives
cνλµ(a, b) =
cνλρ(a, b)− c
λµ(a, b)
|aν | − |aµ|
Set s = r+1. Exactly as in the case |ν|−|µ| = 1, we have the following diagrammatic
relations:
|aρ| − |aµ|
sρ ν = ρ s ν − µ s ν
|aν | − |aρ|
sρ ν = µ r ν − µ r ρ
Hence, the desired formula for cνλµ(a, b) will follow if we prove the relation
µ r ν = µ s ν
We construct a weight-preserving bijection between the barred reverse λ-tableaux
which are represented by the left and right hand sides of this diagrammatic relation.
Here the weight is the product on the right hand side of (2.4) corresponding to a
barred tableau. Let such a tableau with a barred entry r in the box (i, j) be given.
Suppose first that the box (i − 1, j) belongs to the diagram and it is occupied by
s = r+1. Then the image of the tableau under the map is the same tableau but the
entry T (i, j) = r is now unbarred while T (i− 1, j) = r + 1 is barred. Since
(aν)r+1 = (aµ)r and T (i− 1, j)− c(i− 1, j) = T (i, j)− c(i, j),
the weights of the tableaux are preserved under the map.
Suppose now that the entry in the box (i − 1, j) is greater than r + 1, or this
box is outside the diagram. Consider all entries r in the row i to the left of the box
(i, j) and suppose that they occupy the boxes (i, j −m), (i, j −m+1), . . . , (i, j− 1).
Then the image of the tableau under the map is the tableau obtained by replacing
the entries in each of the boxes (i, j −m), . . . , (i, j) with s = r + 1 and barring the
entry in the box (i, j −m). The weights of the tableaux are again preserved.
The inverse map is described in a similar way. This gives the desired weight-
preserving bijection. The general argument uses similar calculations with the barred
diagrams and a similar bijection described in [17].
Remark 2.5. (i) A cohomological interpretation of the coefficients cνλµ(a, b) and their
puzzle computation can be found in [9].
(ii) The definition (2.3) of the coefficients cνλµ(a, b) can be extended to the case
where λ is a skew diagram. Lemma 2.4 and its proof remain valid; see [17].
(iii) In contrast with the Littlewood–Richardson polynomials cνλµ(a), the coeffi-
cients cνλµ(a, b) do not have the stability property as they depend on n.
Lemma 2.4 implies that the Littlewood–Richardson polynomials can be calcu-
lated by (2.4) with b = a, that is, cνλµ(a) = c
λµ(a, a). Our strategy now is to show
that (unlike the formula of Theorem 3.1 in [17]), the formula (2.4) (with b = a) is
“nonnegative” in the sense that all nonzero products which occur in the formula are
polynomials in the ai − aj with i < j. Then we demonstrate that the ν-boundness
condition serves to eliminate the unwanted zero terms.
Lemma 2.6. Let R be a sequence of the form (2.1) and let T ∈ T (λ,R). Suppose
that ∏
T (α) unbarred
aT (α)−ρ(α)
T (α)
− aT (α)−c(α)
6= 0. (2.7)
Then ρ(α)
T (α)
> c(α) for all α ∈ λ with unbarred T (α).
Proof. Suppose on the contrary that there exists a box α = (i, j) with an unbarred
T (i, j) and the condition ρ(i, j)T (i,j) < j − i; the equality ρ(i, j)T (i,j) = j − i is
excluded since this would violate (2.7). Choose such a box with the minimum possible
value of j. If all the entries T (i, 1), . . . , T (i, j − 1) of T are barred then ρ(i, j) is
obtained from µ by adding boxes in rows T (i, 1) > · · · > T (i, j − 1) and, possibly,
by adding other boxes. Since T (i, j − 1) > T (i, j), we have ρ(i, j)T (i,j) > j − 1, a
contradiction. So, at least one of the entries T (i, 1), . . . , T (i, j−1) must be unbarred.
Take such an unbarred entry T (i, k) which is the closest to T (i, j), that is, all entries
T (i, k+1), . . . , T (i, j − 1) are barred. Then ρ(i, j) is obtained from ρ(i, k) by adding
boxes in rows T (i, k + 1) > · · · > T (i, j − 1) and, possibly, by adding other boxes.
Hence,
ρ(i, j)T (i,j) > ρ(i, k)T (i,k) + j − k − 1
which implies ρ(i, k)T (i,k) < k − i + 1. However, if ρ(i, k)T (i,k) = k − i then the
factor in (2.7) corresponding to α = (i, k) is zero, which is impossible. Therefore
ρ(i, k)T (i,k) < k − i which contradicts the choice of j.
Lemma 2.7. Suppose that R is a sequence of the form (2.1) and T ∈ T (λ,R). If
(2.7) holds then T is ν-bounded.
Proof. By Lemma 2.6, for all unbarred entries T (1, k) of the first row of the tableau
T we have ρ(1, k)T (1,k) > k. This implies νT (1,k) > k. If the entry T (1, j) is barred
then ρ(1, k)T (1,k) > k for the nearest unbarred entry T (1, k) on its left (if it exists).
Then ν is obtained from ρ(1, k) by adding boxes in rows T (1, k + 1) > · · · > T (1, j)
and, possibly, by adding other boxes. This implies νT (1,j) > j. Thus, this inequality
holds for all j = 1, . . . , λ1. This is equivalent to the ν-boundness of T .
Lemma 2.8. Suppose that R is a sequence of the form (2.1) and T ∈ T (λ,R) is
ν-bounded. Then ρ(α)
T (α)
> c(α) for all α ∈ λ with unbarred T (α).
Proof. We argue by contradiction. Taking into account Lemma 2.6, we find that for
some α = (i, j) with unbarred T (α) we have ρ(i, j)T (i,j) = j − i. Set t = T (i, j)
and consider all barred entries of T (assuming for now they exist) which are equal
to t and occur to the right of the column j. Since T is a reverse tableau, these
entries t̄ can only occur in rows 1, 2, . . . , i. Let (r, k) be the box with the maximum
column number k containing t̄. Then the total number of such entries t̄ does not
exceed k− j. This implies that the number of boxes νt in row t of ν does not exceed
ρ(i, j)t + k − j = k − i. Hence, ν
k 6 t − 1. On the other hand, by the ν-boundness
of T we have t = T (r, k) 6 T (1, k) 6 ν ′k, a contradiction.
If none of the boxes to the right of the column j contains t̄ then νt = ρ(i, j)t = j−i.
However, by the assumption, νt > νT (1,j) > j, a contradiction.
This completes the proof of the theorem.
By the column word of a tableau T we will mean the sequence of all entries of T
written in the column order.
Corollary 2.9. Suppose that |ν| = |λ| + |µ|. The Littlewood–Richardson coefficient
cνλµ equals the number of ν-bounded reverse λ-tableaux T whose column word coincides
with the Yamanouchi symbol of a certain sequence R of the form (2.1).
This can be shown to be equivalent to a well-known version of the Littlewood–
Richardson rule. Corollary 2.9 also holds with the ν-boundness condition dropped;
see Lemma 2.7. By the corollary, cνλµ counts the cardinality of the intersection of two
finite sets: the set of column words of ν-bounded reverse λ-tableaux and the set of
Yamanouchi symbols of the sequences of the form (2.1).
Remark 2.10. Due to (1.10), the multiplication rule for the polynomials sλ(x|u) is
obtained from Theorem 2.1 by replacing ai with un−i+1 for each i. The corresponding
coefficients are polynomials in the ui−uj, i > j, with positive integer coefficients.
Corollary 2.11. Suppose that the polynomials cνλµ(a) are defined by the expansion
(1.3) with x = (x1, . . . , xn). Then c
λµ(a) is independent of n as soon as n > ν
Moreover, if n < ν ′1 then c
λµ(a) = 0.
Proof. This follows from the boundness condition on the reverse tableaux.
3 Applications
3.1 Equivariant Schubert calculus on the Grassmannian
As in the Introduction, consider the equivariant cohomology ring H∗T (Gr(n,N)) as
a module over Z[t1, . . . , tN ]. Let x1, . . . , xn denote the Chern roots of the dual S
of the tautological subbundle S of the trivial bundle CNGr(n,N) so that for the total
equivariant Chern class of S we have
cT (S) =
(1− xi).
Then, due to [6, Lecture 8, Proposition 1.1] (see also [16]), the equivariant Schubert
classes σλ can be expressed by
σλ = sλ(x|u), u = (−tN , . . . ,−t1, 0, . . . ).
Hence, Theorem 2.1 yields a multiplication rule for the equivariant Schubert classes.
The corresponding stability property is implied by Corollary 2.11.
Corollary 3.1. We have
σλ σµ =
d νλµ σν ,
where
d νλµ =
T (α) unbarred
tm+T (α)−c(α) − tm+T (α)−ρ(α)
T (α)
, (3.1)
summed over all sequences R of the form (2.1) and all ν-bounded reverse λ-tableaux
T ∈ T (λ,R). In particular, the d νλµ are polynomials in the ti− tj, i > j, with positive
integer coefficients. Moreover, the coefficients d νλµ, regarded as polynomials in the
variables ai defined in (1.5), are independent of n and m, as soon as the inequalities
n > λ′1 + µ
1 and m > λ1 + µ1 hold.
Example 3.2. For any n > 3 and m > 4 we have
σ(2) σ(2,1) = σ(4,1) + σ(3,2) + σ(3,1,1) + σ(2,2,1)
+ (tm+2 − tm−1 + tm − tm−2) σ(3,1) + (tm+2 − tm−1) σ(2,2)
+ (tm − tm−1) σ(2,1,1) + (tm+2 − tm−1) (tm − tm−1) σ(2,1).
This follows from Example 2.2.
The first manifestly positive rule for the expansion of σλ σµ was given by Knutson
and Tao [9] by using combinatorics of puzzles. Although the stability property was not
pointed out in [9], it can be deduced directly from the puzzle rule or by applying the
weight-preserving bijection between the puzzles and the barred tableaux constructed
by Kreiman [10].
3.2 Quantum immanants and higher Capelli operators
Let gln denote the general linear Lie algebra over C. Consider the center Z(gln) of
the universal enveloping algebra U(gln). The algebra U(gln) is equipped with the
natural filtration. For all n we identify gln−1 as a subalgebra of gln in a usual way
and denote by gl∞ the corresponding inductive limit
gl∞ =
Due to Olshanski [20], there exist filtration-preserving homomorphisms
on : Z(gln) → Z(gln−1), n > 1, (3.2)
which allow one to define the algebra Z of the virtual Casimir elements for the Lie
algebra gl∞ as the inverse limit
Z = lim
Z(gln), n → ∞,
in the category of filtered algebras.
The quantum immanants Sλ|n are elements of the center Z(gln) of the universal
enveloping algebra U(gln) parameterized by the diagrams λ with at most n rows;
see [18]. The elements Sλ|n form a basis of Z(gln) and they are consistent with the
Olshanski homomorphisms (3.2) so that
on : Sλ|n 7→ Sλ|n−1, (3.3)
where we assume Sλ|n = 0 if the number of rows of λ exceeds n. For any diagram λ,
the corresponding virtual quantum immanant Sλ is then defined as the sequence
Sλ = ( Sλ|n |n > 0).
The elements Sλ parameterized by all diagrams λ form a basis of the algebra Z so
that we can define the coefficients f νλµ by the expansion
Sλ Sµ =
f νλµ Sν .
Note that the same coefficients f νλµ determine the multiplication rule for the higher
Capelli operators ∆λ, which are defined as the sequences of the images of the quantum
immanants Sλ|n, where each image is taken under a natural representation of gln by
differential operators; see [18, 19].
Corollary 3.3. The coefficient f νλµ is zero unless µ ⊆ ν. If µ ⊆ ν then
f νλµ =
T (α) unbarred
ρ(α)T (α) − c(α)
, (3.4)
summed over all sequences R of the form (2.1) and all ν-bounded reverse λ-tableaux
T ∈ T (λ,R). In particular, the f νλµ are nonnegative integers.
Proof. Due to the stability property (3.3) of the quantum immanants, it suffices to
calculate the corresponding coefficients for the expansion of the products Sλ|n Sµ|n.
The images of the quantum immanants Sλ|n under the Harish-Chandra isomorphism
can be identified with the double Schur polynomials sλ(x||a) where the sequence a is
specialized to ai = −i; see [18]. Therefore, the coefficients in question coincide with
the corresponding specializations of the Littlewood–Richardson polynomials cνλµ(a).
Example 3.4. Using Example 2.2 we get
S(2) S(2,1) = S(4,1) + S(3,2) + S(3,1,1) + S(2,2,1) + 5 S(3,1) + 3 S(2,2) + S(2,1,1) + 3 S(2,1).
In the course of the proof of Corollary 3.3 we also calculated the coefficients
for the expansion of the products Sλ|n Sµ|n for any n. Some other formulas for these
coefficients were obtained in [17]. In particular, it was shown that the f νλµ are integers,
although their positivity property was not established there.
Note also that the algebra of virtual Casimir elements Z is isomorphic to the
algebra of shifted symmetric functions Λ∗; see [19]. The latter can be regarded as the
specialization of Λ (or rather, its extension over C) at ai = −i for all i ∈ Z.
4 Supertableau formulas for sλ(x||a) and c
λµ(a)
Here we obtain one more rule for the calculation of the Littlewood–Richardson poly-
nomials cνλµ(a). It relies on a supertableau representation of the double Schur poly-
nomials sλ(x||a) which is implied by the results of [13]. This representation provides
a “finite” version of the supertableau formulas of [7] and [13]; cf. [4].
Fix a positive integer n. For r > 1 set u(r) = (u1, . . . , ur) and use the 9th Variation
in [13] with the indeterminates hrs specialized by
hrs = hr(u
(n−r−s+1)) if r + s 6 n,
and 0 otherwise, where hr denotes the r-th complete symmetric polynomial. Let us
write ŝλ/µ(u) for the corresponding Schur functions. Then (8.2) and (9.1) in [13] give
ŝλ/µ(u) =
α∈λ/µ
uT (α),
summed over semistandard tableaux T of shape λ/µ, such that the entries of the i-th
row do not exceed n− λi + i. Furthermore, using (6.18)
1 and (9.6 ′) in [13] we get
sλ(x|u) =
sµ(x) ŝλ′/µ′(−u). (4.5)
Equivalently, this can be interpreted as a combinatorial expression for the polynomials
sλ(x|u) in terms of “supertableaux”. Identify the indices of u with the symbols
1′, 2′, . . . . A supertableau T is obtained by filling in the diagram of λ with the indices
1, . . . , n, 1′, 2′, . . . in such a way that in each row (resp. column) each primed index is
to the right (resp. below) of each unprimed index; unprimed indices weakly increase
along the rows and strictly increase down the columns; primed indices strictly increase
along the rows and weakly increase down the columns; primed indices in column j
do not exceed n− λ′j + j. Relation (4.5) implies the following.
Proposition 4.1. We have
sλ(x|u) =
T (α) unprimed
xT (α)
T (α) primed
(−uT (α)), (4.6)
summed over all λ-supertableaux T .
Using (1.10), we get an analogous representation for the double Schur polynomials
sλ(x||a). A reverse supertableau T is obtained by filling in the diagram of λ with
1This formula in [13] should be corrected by replacing a(λj+n−j) with a(λi+n−i).
the indices 1, . . . , n, n′, (n − 1)′, . . . (including non-positive primed indices) in such
a way that in each row (resp. column) each primed index is to the right (resp.
below) of each unprimed index; unprimed indices weakly decrease along the rows and
strictly decrease down the columns; primed indices strictly decrease along the rows
and weakly decrease down the columns; primed indices in column j are not less than
λ′j − j + 1. The following supertableau representation of the polynomials sλ(x||a)
follows from Proposition 4.1.
Corollary 4.2. We have
sλ(x||a) =
T (α) unprimed
xT (α)
T (α) primed
(−aT (α)), (4.7)
summed over all reverse λ-supertableaux T .
Example 4.3. Let n = 2 and λ = (2, 1). By the definition (1.9),
s(2,1)(x||a) = (x2 − a2)(x1 − a0)(x1 − a2) + (x2 − a2)(x2 − a1)(x1 − a2).
On the other hand, the reverse (2, 1)-supertableaux are
2 0 ′
2 1 ′
2 2 ′
2 0 ′
2 1 ′
2 2 ′
1 0 ′
1 1 ′
1 2 ′
2 ′ 0 ′
2 ′ 1 ′
which yield
s(2,1)(x||a) = x
1x2 + x1x
2 − x1x2a2 − x
2a2 − x
1a2 − x1x2a0 − x1x2a1 − x1x2a2
+ x2a0a2 + x2a1a2 + x2a
2 + x1a0a2 + x1a1a2 + x1a
2 − a0a
2 − a1a
Formula (4.5) implies a supertableau representation of the coefficients cνλµ(a, b)
and hence, of the Littlewood–Richardson polynomials cνλµ(a). The representation
for the latter is neither manifestly positive, nor stable; it provides an expression for
cνλµ(a) as an alternating sum of monomials in the ai. Given a sequence R of the form
(2.1), construct the set S(λ,R) of barred reverse λ-supertableaux by analogy with
T (λ,R). A tableau T ∈ S(λ,R) must contain boxes α1, . . . , αl occupied by unprimed
indices r1, r2, . . . , rl listed in the column order which is restricted to the subtableau of
T formed by the unprimed indices. As before, we distinguish the entries in α1, . . . , αl
by barring each of them. For each box α with αi ≺ α ≺ αi+1, 0 6 i 6 l, which is
occupied by an unprimed index, set ρ(α) = ρ(i).
Corollary 4.4. The coefficients cνλµ(a, b) defined in (2.3) can be given by
cνλµ(a, b) =
T (α) unprimed, unbarred
aT (α)−ρ(α)
T (α)
T (α) primed
(−bT (α)), (4.8)
summed over sequences R of the form (2.1) and reverse supertableaux T ∈ S(λ,R).
Proof. Applying formula (4.5) we can reduce the calculation of cνλµ(a, b) to the par-
ticular case of the sequence b = (0). Now (4.8) follows from Lemma 2.4.
Example 4.5. In order to calculate the Littlewood–Richardson polynomial c
(2,1)
(2,1) (2)
we may take n = 2; see Corollary 2.11. The barred reverse supertableaux compatible
with the sequence (2) → (2, 1) are
2 0 ′
2 1 ′
2 2 ′
2 0 ′
2 1 ′
2 2 ′
so that
(2,1)
(2,1) (2)
(a) = a2−1 + a−1a1 + a−1a2 − a−1a2 − a1a2 − a
− a−1a0 − a−1a1 − a−1a2 + a0a2 + a1a2 + a
= a2−1 − a−1a0 − a−1a2 + a0a2,
which agrees with Example 2.2.
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	Introduction
	Multiplication rule
	Applications
	Equivariant Schubert calculus on the Grassmannian
	Quantum immanants and higher Capelli operators
	Supertableau formulas for s(x || a) and c(a)
