JACOBI-TRUDI TYPE FORMULA FOR PARABOLICALLY SEMISTANDARD TABLEAUX

Abstract

The notion of a parabolically semistandard tableau is a generalisation of Young tableau, which explains combinatorial aspect of various Howe dualities of type A. We prove a Jacobi-Trudi type formula for the character of parabolically semistandard tableaux of a given generalised partition shape by using non-intersecting lattice paths.

1. INTRODUCTION

A Schur polynomial is a symmetric polynomial, which plays an important role in algebraic combinatorics and representation theory (we refer the reader to [6, 15, 16] for general exposition on Schur polynomials). Let x1, . . . , xn be mutually commuting n variables. Let sλ(x1, . . . , xn) be the Schur polynomial corresponding to a partition λ = (λ1, . . . , λn). It is well-known that sλ(x1, . . . , xn) is the character of a complex irreducible polynomial representation of the general linear group GLn(ℂ) whose highest weight corresponds to λ. There are several equivalent definitions of sλ(x1, . . . , xn). One is the celebrated Weyl character formula, which has been extended to the case of a symmetrizable Kac-Moody algebra [9]. There is a combinatorial formula, where sλ(x1, . . . , xn) is given as the weight generating function of the set of Young tableaux of shape λ. Another well-known one is the Jacobi-Trudi formula

where hk(x1, . . . , xn) is the kth complete symmetric polynomial. While the above formula is originally due to Jacobi, Gessel and Viennot introduced a new interesting proof in terms of non-intersecting lattice paths [7], which has resulted in various generalizations and applications in combinatorics.

In [12], Kwon introduced a new combinatorial object, which we call parabolically semistandard tableaux, in order to understand the combinatorial aspect of Howe duality of type A [8]. For a generalized partition λ of length n, the weight generating function Sλ of parabolically semistandard tableaux of shape λ gives the character of an irreducible representation of a general linear Lie (super)algebra 𝔤, which arises from (𝔤, GLn(ℂ)-duality on various Fock spaces. The character Sλ includes a usual Schur polynomial as a special case, and it also has a Weyl-Kac type character formula, and a Jacobi-Trudi type formula (see also [13]).

The goal of this paper is to show a Jacobi-Trudi type formula for Sλ by using non-intersecting lattice paths. Since a parabolically semistandard tableau is roughly speaking a pair (S, T) of skew-shaped Young tableaux with a common inner shape and each component corresponds to an n-tuple of non-intersecting lattice paths, the pair (S, T) corresponds to an n-tuple of non-intersecting zigzag-shaped lattice paths which is obtained by gluing non-intersecting paths associated to S and T. This is our key observation. Then we apply the arguments similar to [7] to obtain a Jacobi-Trudi type formula for Sλ.

 

2. PARABOLICALLY SEMISTANDARD TABLEAUX

2.1. Young tableaux Let us briefly recall necessary background on Young tableaux (see [6] for more details). We denote by ℤ and ℤ>0 the set of integers and positive integers, respectively. A partition is a weakly decreasing sequence of non-negative integers λ = (λ1, λ2, . . .) such that Σi≥1 λi is finite. We say that λ is a partition of n if Σi≥1 λi = n and denote by ℓ(λ) the number of positive entries of λ. Let be the set of all partitions, and put for n ≥ 1.

A Young diagram is a collection of boxes arranged in left-justified row, with weakly decreasing number of boxes in each row from top to bottom. A Young diagram determines a unique partition λ = (λ1, λ2, · · ·), where λi is the number of boxes in the ith row of the diagram. From now on, we identify a Young diagram with its partition.

Example 2.1. The Young diagram corresponding to the partition λ = (5, 3, 3, 1) is

Let be given. A Young tableau T is a filling of λ or the boxes in its Young diagram with positive integers such that the entries are weakly increasing from left to right in each row, and strictly increasing from top to bottom in each column. We say that λ is the shape of T, and write sh(T) = λ.

Example 2.2. For λ = (5, 3, 3, 1)

is a Young tableau of shape λ.

For with λ ⊃ µ (that is, λi ≥ µi for all i), λ/µ denotes the skew Young diagram. A skew Young tableau is a filling of a skew Young diagram λ/µ with positive integers in the same way as in the case of Young tableaux.

Example 2.3. For λ/µ = (5, 3, 3, 1)/(2, 1),

is a skew Young tableau of shape λ/µ.

Let x = {x1, x2, . . .} be a set of formal commuting variables. For a Young tableau T, we put , where mi is the number of times i occurs in T. For T in Example 2.2, we have . Let sλ(x) = ΣTxT be the Schur function corresponding to , where the sum is over all Young tableaux T of sh(T) = λ. For k ≥ 0, let hk(x) = s(k) (x), which is called the kth complete symmetric function. For , we put hµ(x) = hµ1(x)hµ2(x). . ..

There is another well-known equivalent definition of a Schur function called the Jacobi-Trudi formula, which expresses a Schur function as a determinant, and hence as a linear combination of hµ(x)’s for (cf. [6]).

Theorem 2.4. For with ℓ(λ) ≤ n,

where we assume that h−k(x) = 0 for k ≥ 1.

2.2. Parabolically semistandard tableaux Let 𝓐 be a linearly ordered countable set with a ℤ2-grading 𝓐 = 𝓐0 ⨆ 𝓐1. For a ∈ 𝓐, a is called even (resp. odd) if a ∈ 𝓐0 (resp. a ∈ 𝓐1). Let λ/µ be a skew Young diagram. A tableau T obtained by filling λ/µ with entries in 𝓐 is called 𝓐-semistandard if the entries in each row (resp. column) are weakly increasing from left to right (resp. from top to bottom), and the entries in 𝓐0 (resp. 𝓐1) are strictly increasing in each column (resp. row). We say that λ/µ is the shape of T, and write sh(T) = λ/µ. We denote by SST𝓐(λ/µ) the set of all 𝓐-semistandard tableaux of shape λ/µ. We set . Let x𝓐 = { xa | a ∈ 𝓐 } be a set of formal commuting variables indexed by 𝓐. For T ∈ SST𝓐(λ/µ), put , where ma is the number of occurrences of a in T. We define the character of SST𝓐(λ/µ) to be .

We assume that ℤ>0 is given with a usual linear ordering and all entries even. When 𝓐 = ℤ>0, an 𝓐-semistandard tableau is a (skew) Young tableau, and sλ(x𝓐) is the Schur function associated to .

Let be the set of all generalized partitions of length n. We may identify λ with a generalized Young diagram as in the following example.

Example 2.5. The generalized partition corresponds to

Suppose that 𝓐 and 𝓑 are two disjoint linearly ordered ℤ2-graded countable sets. Now, let us introduce our main combinatorial object.

Definition 2.6 ([12]). For , a parabolically semistandard tableau of shape λ (with respect to (𝓐, 𝓑)) is a pair of tableaux (T+, T−) such that

for some integer d ≥ 0 and satisfying (1) , and (2) µ ⊂ (dn), µ ⊂ λ+(dn). We denote by SST𝓐/𝓑(λ) the set of all parabolically semistandard tableaux of shape λ with respect to (𝓐, 𝓑).

Roughly speaking, a parabolically semistandard tableau of shape λ is a pair of 𝓐-semistandard tableau and 𝓑-semistandard tableau whose shapes are not necessarily fixed ones but satisfy certain conditions determined by λ.

Example 2.7. Suppose that 𝓐 = ℤ>0 = { 1 < 2 < 3 < . . . } and 𝓑 = ℤ<0 = { −1 < −2 < −3 < . . . } with all entries even. Then

where the vertical lines in T+ and T− correspond to the one in the generalized partition (3, 2, 0, −2). In this case, we have sh(T+) = ((3, 2, 0, −2) + (34))/(2, 1, 0, 0), and sh(T−) = (34) / (2, 1, 0, 0).

For , we define the character of SST𝓐/𝓑(λ) to be

We put . For k ∈ ℤ, we put .

2.3. Irreducible characters Let us briefly recall a representation theoretic meaning of parabolically semistandard tableaux. For an arbitrary ℤ2-graded linearly ordered set 𝓒, we denote by V𝓒 a superspace with basis { vc | c ∈ 𝓒 }, and let 𝔤𝔩(V𝓒) be the general linear Lie superalgebra spanned by Ecc' for c, c' ∈ 𝓒. Here Ecc' is the matrix where the entry at (c, c')-position is 1 and 0 elsewhere.

Let 𝔤 = 𝔤𝔩(V𝓒) with 𝓒 = 𝓑 ∗ 𝓐, where 𝓑 ∗ 𝓐 is the ℤ2-graded set 𝓐 ⨆ 𝓑 with the extended linear ordering defined by y < x for all x ∈ 𝓐 and y ∈ 𝓑. Let

be the super symmetric algebra generated by , where is the restricted dual space of V𝓑. One can define a semisimple action of 𝔤 on , and a semisimple action of GLn(ℂ) on for n ≥ 1 so that we have the following multiplicity-free decomposition as a (𝔤, GLn(ℂ))-module,

for a subset Hn of , where Ln(λ) is the irreducible GLn(ℂ)-module with highest weight λ ∈ Hn, and L(λ) is an irreducible 𝔤-module corresponding to Ln(λ) (see the arguments in [4, Sections 5.1 and 5.4]). We define the character chL(λ) to be the trace of the operator on L(λ) for λ ∈ Hn. Finally from a Cauchy type identity for parabolically semistandard tableaux [12, Theorem 4.1], we can conclude the following (cf. [14, Theorem 2.3]).

Theorem 2.8. For n ≥ 1, we have

as a (𝔤, GLn(ℂ))-module, that is, , and the irreducible character chL(λ) is given by for .

Recall that when 𝓐 is finite with 𝓐 = 𝓐0 or 𝓐1 and , the decomposition in Theorem 2.8 is the classical (GLℓ(ℂ), GLn(ℂ))-Howe duality on the symmetric algebra or exterior algebra generated by ℂℓ ⊗ ℂn, where ℓ = |𝓐| (cf. [8]). Moreover, the decomposition in Theorem 2.8 includes other Howe dualities of type A which have been studied in [1, 2, 3, 5, 8, 10, 11] under suitable choices of 𝓐 and 𝓑 (see [12] for more details).

 

3. JACOBI-TRUDI FORMULA

3.1. Lattice paths

Definition 3.1. A lattice path is a sequence

of points v1, ..., vr in ℤ × ℤ with vi = (ai, bi) such that b1 < 0 < br, and

Example 3.2. The following path

is the lattice path

We denote by 𝒫 the set of lattice paths. Let p = v1...vr ∈ 𝒫 be given with vi = (ai, bi) for 1 ≤ i ≤ r. We often identify p with its extended lattice path v0v1...vrvr+1, where v0 = (a1, −∞) and vr+1 = (ar, ∞). Here we regard (a1, −∞) as a point below (a1, y) for all y ≤ b1, and (ar, ∞) as a point above (ar, y) for all y ≥ br. We also write p : v0 → vr+1. For 0 ≤ i ≤ r, let vivi+1 denote the line segment joining vi and vi+1, where we understand v0v1 (resp. vrvr+1) as an half-infinite line joining (a1, b1) and (a1, −∞) (resp. (ar, br) and (ar, ∞)). Let z = { zi | i ∈ ℤ× } be a set of formal commuting variables, where ℤ× = ℤ \ {0}. We consider a weight monomial

Example 3.3. For a lattice path

its weight monomial is (the numbers on the horizontal line segments denote their y-coordinates in ℤ × ℤ).

Fix a positive integer n. Let Sn be the group of permutations on n letters. Let α = (α1, . . . , αn), β = (β1, . . . , βn) ∈ ℤn be given with α1 > . . . > αn and β1 > . . . > βn. We define

Put , and (−1)p = sgn(π) for p ∈ 𝒫 (α, β) with its associated permutation π ∈ Sn.

Example 3.4. Let n = 4, α = (1, 0, −1, −2) and β = (4, 2, −1, −4). Then

with the associated permutation

A weight monomial of p is

and (−1)p = sgn(π) = 1.

Let us define a map

as follows; for p = (p1, ..., pn) ∈ 𝒫(α, β)

(1) If for all 1 ≤ i ≠ j ≤ n, then ϕ(p) = p. (2) Otherwise, we choose the largest i such that pi has an intersection point w with pj for some i > j, and assume that w is the first intersection point appearing in pi from the bottom. Then we define ϕ(p) to be the n-tuple of paths obtained from p by replacing

Example 3.5. Let p be as in Example 3.4. Then

By definition of ϕ , we can check that for p ∈ 𝒫(α, β)

(1) ϕ(p) = p if and only if p has no intersection point, (2) ϕ2(p) = p, (3) zϕ(p) = zp, (4) (−1)ϕ(p) = −(−1)p.

We put

the set of fixed points in 𝒫(α, β) under ϕ, or the subset of p in 𝒫(α, β) with no intersection point.

For , we define

where δ = (0, −1, . . . , −n + 1) and λ + δ = (λ1, λ2 − 1, . . . , λn − n + 1). For k ∈ ℤ, we put 𝘚k = 𝘚(k). Then we have the following Jacobi-Trudi formula for 𝘚λ, which is an analogue of [12] for our zigzag-shaped lattice paths.

Proposition 3.6. For , we have

Proof. Recall from (4) that for 1 ≤ i, j ≤ n

where we shift the x-coordinates in p by −j + 1. Thus

Since ϕ(p) ≠ p for p ∉ 𝒫0(δ, λ + δ) and (−1)ϕ(p) = −(−1)p, we have

Also note that (−1)p = 1 for p ∈ 𝒫0(δ, λ + δ). Therefore, we have

                              ☐

3.2. Non-intersecting paths and Young tableaux Let α = (α1, . . . , αn), be such that α1 > . . . > αn, β1 > . . . > βn and αi ≤ βi for all i. Consider an n-tuple p = (p1, . . . , pn) of non-intersecting (extended) lattice paths where

for 1 ≤ i ≤ n. Note that pi is a lattice path starting from a point (αi, 0), which is a upper half of a lattice path defined in Definition 3.1. Put δ = (0, −1, . . . , −n + 1). Choose d ≥ 0 satisfying

If we put µ = α − δ + (dn) and λ = β − δ + (dn), then λ/µ is a skew Young diagram.

Now, associated to p, we define a tableau T of shape λ/µ with entries in ℤ>0 as follows. For 1 ≤ i ≤ n with αi < βi and 1 ≤ j ≤ βi − αi, we fill the box in the ith row and jth column of λ/µ with k if

The following lemma is well-known [7]. But we give a detailed proof for the readers’ convenience.

Lemma 3.7. Under the above assumptions, T is ℤ>0-semistandard or a Young tableau of shape λ/µ.

Proof. Fix 1 ≤ i ≤ n. Let Ti,j denote the jth (non-empty) entry of T (from the left) in the ith row (from the top) for 1 ≤ j ≤ βi − αi.

It is clear that the entries of T in each row are weakly increasing from left to right since the y-coordinates of each path pi : (αi, 0) → (βi,∞) are weakly increasing from bottom to top. Hence it is enough to show that the entries of T in each column are strictly increasing from top to bottom.

Fix 1 ≤ i < n. Suppose first that αi − αi+1 = ℓ ≥ 1. Then µi − µi+1 = {αi −(−i+ 1) +d} − {αi −(−i) +d} = (αi −αi+1)−1 = ℓ−1. This implies that Ti,j and Ti+1,j+(ℓ−1) are in the same column in T for all j such that Ti,j and Ti+1,j+(ℓ−1) are non-empty. The jth and (j + ℓ − 1)th horizontal line segments of pi and pi+1 are given by

respectively, where k = Ti,j and k' = Ti+1,j+(ℓ−1). If k ≥ k', then the paths pi and pi+1 necessarily have an intersection point, which is a contradiction. Therefore, Ti,j < Ti+1,j+(ℓ−1).                               ☐

Note that the shape of T does not depend on the choice of d, and the correspondence p ↦ T gives a bijection between the set of non-intersecting paths satisfying (5) and SSTℤ>0 (λ/µ).

Example 3.8. Consider a quadruple of non-intersecting paths p = (p1, p2, p3, p4) with α = (−1, −3, −5, −6) and β = (3, 1, −2, −5)

Then the associated Young tableau is

Now, consider parabolically semistandard tableaux, where 𝓐 = ℤ>0 = { 1 < 2 < 3 < . . . } and 𝓑 = ℤ<0 = { −1 < −2 < −3 < . . . } with all entries even. Note that the linear ordering on 𝓑 is a reverse ordering of the usual one. Then we have

Proposition 3.9. For , there exists a bijection

Proof. Let p = (p1, . . . , pn) ∈ 𝒫0(δ, λ + δ) be given with

for some γi ∈ ℤ (1 ≤ i ≤ n). Then we put , where , an upper half of pi with the vertices having non-negative second components, and put , where , the lower half of pi. .

Choose d ≥ 0 such that γ − δ + (dn), . First, as in Lemma 3.7, we may associate a Young tableau T+ of shape (λ + (dn))/µ where µ = γ − δ + (dn).

Let , where is obtained by reversing the order of the vertices in and changing the sign of their second components. By the same argument, we may associate a Young tableau of shape (dn)/µ, and then replace an entry k with −k once again to get a ℤ<0-semistandard tableau T− of (dn)/µ.

We define a map ψ : 𝒫0(δ, λ + δ) → SSTℤ>0/ℤ<0(λ) by ψ(p) = (T+, T −). Since the correspondence p ↦ (T+, T−) is reversible, ψ is a bijection.                               ☐

Remark 3.10. The bijection ψ in Proposition 3.9 preserves weight in the following sense: If (T+, T−) = ψ(p) for p ∈ 𝒫0(δ, λ + δ), then , where we assume that zk = xk and for k ≥ 1.

Example 3.11. Let p ∈ 𝒫0(δ, λ + δ) be a 4-tuple of lattice paths with δ = (0, −1, −2, −3) and λ + δ = (3, 1, −2, −5) as follows.

Then

Now, we are in a position to prove our main theorem.

Theorem 3.12. For , we have

Proof. Let us assume that zk = xk and for k ≥ 1. Then we have for l ∈ ℤ. So by Proposition 3.6, we have

On the other hand, by Proposition 3.9 and Remark 3.10 we have

Combining (6) and (7), we obtain

This completes the proof.                               ☐

3.3. General cases for 𝓐 and 𝓑 In this subsection, we prove that Theorem 3.12 can be naturally extended to the case of , where 𝓐 = ℤ>0 = { 1 < 2 < 3 < . . . } and 𝓑 = ℤ<0 = { −1 < −2 < −3 < . . . } with arbitrary ℤ2-gradings.

For this, we consider a lattice path p = v1...vr of points v1, ..., vr in ℤ × ℤ with vi = (si, ti) satisfying the following conditions:

(1) t1 < 0 < tr, (2) if ti ≠ 0 and ti ∈ 𝓐0 ⨆ 𝓑0, then (3) if ti ≠ 0 and ti ∈ 𝓐1 ⨆ 𝓑1, then (4) if ti = 0, then vi+1 − vi = (0, 1).

We may define the notion of an extended path in the same way as in Section 3.1, and accordingly 𝒫(α, β), the involution ϕ, and 𝒫0(α, β). For an (extended) path p and z = { zi | i ∈ ℤ× } the set of formal commuting variables, we put

where p = (s1, −∞)v1 . . . vr(sr,∞) with vi = (si, ti) for 1 ≤ i ≤ r. Then we define

and we have

by the same arguments as in Proposition 3.6.

Example 3.13. Suppose that ℤ2-gradings on 𝓐 and 𝓑 are given by

For a lattice path

its weight monomial is (the numbers on the horizontal or the diagonal denote their y-coordinates in ℤ × ℤ).

Now, for , there is also a weight-preserving bijection from 𝒫0(δ, λ + δ) to SST𝓐/𝓑(λ) (see Lemma 3.7 and Proposition 3.9).

Example 3.14. We assume that 𝓐 and 𝓑 are as in (9). Let p ∈ 𝒫0(δ, λ + δ) be a 4-tuple of lattice paths with δ = (0, −1, −2, −3) and λ+δ = (3, 1, −2, −5) as follows.

Then it corresponds to

Therefore, combining with (8), we obtain the Jacobi-Trudi type formula for .

Theorem 3.15. For , we have

Remark 3.16. One can also prove Theorem 3.15 when 𝓐 and 𝓑 are arbitrary two disjoint linearly ordered ℤ2-graded sets, by slightly modifying the notion of extended paths.