Some new identities for Schur functions

Frédéric Jouhet and Jiang Zeng

Institut Girard Desargues, Université Claude Bernard (Lyon 1) 43, bd du 11 Novembre 1918, 69622 Villeurbanne Cedex, France

    jouhet@desargues.univ-lyon1.fr, zeng@desargues.univ-lyon1.fr

Dedicated to Dominique Foata on the occasion of his 65th birthday

Abstract Some new identities for Schur functions are proved. In particular, we settle in the affirmative a recent conjecture of Ishikawa-Wakayama [6] and solve a problem raised by Bressoud [2].

1 Introduction

We fix a positive integer n and let X = (x_{1}, …, x_{n}) be a set of n independent variables. For each partition λ = (λ_{1} ≥ λ_{2} ≥ ⋯ ≥ λ_{n} ≥ 0) of length ≤ n, the Schur function s_{λ}(X) are usually defined as follows [7]:

s_{λ}(X) = det (x_{i}^{λ_{j}+n−j})_{1≤i,j≤n} / det (x_{i}^{n−j})_{1≤i,j≤n} .

In this paper we shall follow the standard definitions and notations of Macdonald’s book [7]. Thus the Ferrers diagram of λ is the subset {(i, j)|j ≥ 1, i ≤ λ_{j}} of ℕ^{2}. If the diagram of μ is included in that of λ we note μ ⊆ λ and the skew diagram λ/μ is called a horizontal strip (or h.s. for short) if there is at most one cell in each column of λ/μ. For any partition λ we note c_{j} := c_{j}(λ) the number of columns of length j in λ, i.e. c_{j} = λ_{j} − λ_{j+1} and define

f_{λ}(a, b) = ∏_{j odd} \frac{a^{c_{j}+1} − b^{c_{j}+1}}{a − b} ∏_{j even} \frac{1 − (ab)^{c_{j}+1}}{1 − ab} .

Since f_{λ}(a, 0) = a^{c(λ)}, where c(λ) is the number of columns of odd length of λ, a classical identity of Littlewood [7] reads then as follows :

∑_{λ} f_{λ}(a, 0)s_{λ}(X) = ∏_{i}(1 − ax_{i})^{−1} ∏_{j<k}(1 − x_{j}x_{k})^{−1}.  (1)

Set

Φ(X; a, b) := ∏_{i}(1 − ax_{i})^{−1}(1 − bx_{i})^{−1} ∏_{j<k}(1 − x_{j}x_{k})^{−1}.

In a recent paper [6], Ishikawa and Wakayama gave the following extension of (1):

∑_{λ} f_{λ}(a, b) s_{λ}(X) = Φ(X; a, b).  (2)

As pointed by Bressoud [2], when (a, b) = (1, 0), (1, −1) and (0, 0), identity (2) reduces to the following interesting known identities respectively:

∑_{λ} s_{λ}(X) = ∏_{i=1}^{n} \frac{1}{1 − x_{i}} ∏_{1≤i<j≤n} \frac{1}{1 − x_{i}x_{j}} ,  (3)

∑_{λeven} s_{λ}(X) = ∏_{i=1}^{n} \frac{1}{1 − x_{i}^{2}} ∏_{1≤i<j≤n} \frac{1}{1 − x_{i}x_{j}} ,  (4)

∑_{λ′ even} s_{λ}(X) = ∏_{1≤i<j≤n} \frac{1}{1 − x_{i}x_{j}} ,  (5)

where λ′ is the conjugate of λ.

In this paper we shall give two generalizations of Ishikawa and Wakayama’s formula (2). To state them we need some definitions.

For r ≥ 0, let h_{r}(X) (resp. e_{r}(X)) be the homogeneous (resp. elementary) symmetric function of X and set

P_{r}(a, b, c) = ∑_{k=0}^{r} \frac{a^{k+1} − b^{k+1}}{a − b} \frac{1 − (ab)^{r−k+1}}{1 − ab} c^{k},

Q_{r}(a, b, c) = ∑_{k=0}^{r} h_{r−k}(a, b, c)(abc)^{k}.

For any positive integer sequence ξ = (ξ_{1}, …, ξ_{r}, …), where ξ_{r} ≠ 0 for only a finite number of integers r, set

F_{ξ}(a, b, c) = h_{ξ_{1}}(a, b, c) ∏_{k≥1} P_{ξ_{2k}}(a, b, c)Q_{ξ_{2k+1}}(a, b, c).

For any integer i ≥ 1, let ε_{i} be the i^{th} vector of the canonical basis of ℤ^{∞} and introduce the operator δ_{i} : δ_{i}ξ = ξ − ε_{i} − ε_{i+1} for ξ ∈ ℕ^{∞}. Set δ_{i}F_{ξ}(a, b, c) = F_{δ_{i}ξ}(a, b, c), where P_{k} = Q_{k} = 0 if k < 0 by convention. Hence, to any partition λ of length ≤ n we can associate the polynomial

f_{λ}(a, b, c) := ∑_{k=0}^{n}(−abc)^{k} ∑_{i_{1}<···<i_{k}} δ_{i_{1}} ⋯ δ_{i_{k}}F_{Γ(λ)}(a, b, c),

where Γ(λ) = (c_{1}, c_{2}, …) is the sequence of the multiplicities of parts in the dual of λ, or c_{j} is the number of columns of length j in λ.

Now we can state our first generalization of (2), which gives in fact a positive answer to a conjecture of Ishikawa and Wakayama [6].

Theorem 1 We have

∑_{λ} f_{λ}(a, b, c)s_{λ}(X) = Φ(X; a, b) ∏_{i}(1 − cx_{i})^{−1}.

On the other hand, Macdonald [7, p. 83-84], Désarménien-Stembridge [4, 9] and Okada [8] have given bounded versions of identities (3)-(5), respectively, as follows :

Theorem 2 (Macdonald) For non negative integers m and n,

∑_{λ_{1}≤m} s_{λ}(X) = \frac{det (x_{i}^{j−1} − x_{i}^{m+2n−j})}{∏_{i=1}^{n}(1 − x_{i}) ∏_{i<j}(x_{i} − x_{j})(x_{i}x_{j} − 1)}.

Theorem 3 (Désarménien-Stembridge) For non negative integers m and n,

∑_{{λ_{1}≤2m \atop λ even}} s_{λ}(X) = \frac{det (x_{i}^{j−1} − x_{i}^{2m+2n+1−j})}{∏_{i=1}^{n}(1 − x_{i}^{2}) ∏_{i<j}(x_{i}x_{j} − 1)(x_{i} − x_{j})}.

Remark. This result follows immediately from Macdonald’s formula. Indeed Pieri’s formula implies:

∑_{k=0}^{n} e_{k}(X) ∑_{{λ_{1}≤2m \atop λeven}} s_{λ}(X) = ∑_{λ_{1}≤2m+1} s_{λ}(X).

Since ∑_{k=0}^{n} e_{k}(X) = ∏_{i=1}^{n}(1+x_{i}), we get immediately theorem 3 by applying Macdonald’s formula.

Theorem 4 (Okada) For non negative integers m and n which is even,

∑_{{λ_{1}≤m \atop λ′ even}} s_{λ}(X) = \frac{1}{2} \frac{det (x_{i}^{j−1} − x_{i}^{m+2n−1−j}) + det (x_{i}^{j−1} + x_{i}^{m+2n−1−j})}{∏_{i<j}(x_{i}x_{j} − 1)(x_{i} − x_{j})}.

After giving elementary proofs of (2) and of the last three identities [1, 2], Bressoud [2] raised the problem of finding an extension of (2) for bounded partitions. Our second generalization of (2) will give an answer to Bressoud’s problem [2].

For any sequence ξ ∈ {±1}^{n}, we denote by |ξ|_{−1} the number of −1’s in the sequence ξ, set X^{ξ} = {x_{1}^{ξ_{1}}, …, x_{n}^{ξ_{n}}} and

D(ξ, z) = 1 − z ∏_{i} x_{i}^{(ξ_{i}−1)/2}.

Theorem 5 For non negative integers m and n,

∑_{λ⊆(m^{n})} f_{λ}(a, b)s_{λ}(X) = ∑_{ξ∈{±1}^{n}} β(ξ, a, b)Φ(X^{ξ}; a, b) ∏_{i} x_{i}^{m(1−ξ_{i})/2}

where the coefficient β(ξ, a, b) is equal to

(\frac{a^{m+1}}{D(ξ, 1/a)} − \frac{b^{m+1}}{D(ξ, 1/b)}) \frac{D(ξ, a)D(ξ, b)}{a − b} if |ξ|_{−1} odd,

(\frac{1}{D(ξ, 1)} − \frac{(ab)^{m+1}}{D(ξ, 1/ab)}) \frac{D(ξ, 1)D(ξ, ab)}{1 − ab} if |ξ|_{−1} even.

Remark. Assume that |a| < 1, |b| < 1 and |x_{i}| < 1 (1 ≤ i ≤ n) and let m → ∞, then all the summands tend to 0 except the one corresponding to |ξ|_{−1} = 0, which tends to Φ(X; a, b). Therefore theorem 5 reduces to (2) when m → ∞.

Note also that the special b = 0 case of theorem 5 was proved by Goulden [5].

We shall give the proof of theorem 1 in section 2 and that of theorem 5 in section 3 using Macdonald’s approach [7]. Finally, in section 4, we will show that when (a, b) = (1, 0), (1, −1) and (0, 0), theorem 5 reduces actually to the above results of Macdonald, Désarménien-Stembridge and Okada respectively.

2 Proof of theorem 1

Let P be the set of partitions of length ≤ n. Given a partition λ ∈ P, we denote by H(λ) the set of partitions μ ∈ P such that λ/μ is a horizontal strip. As noticed at the end of [6], identity (2) can be derived from Littlewood’s formula (1) and the so-called Pieri formula (see [7]):

s_{μ}(X) h_{k}(X) = ∑_{{λ:μ∈H(λ) \atop |λ/μ|=k}} s_{λ}(X).  (6)

In the same vain, we shall derive theorem 1 from (2) and (6). We first review such a proof for (2). By virtue of (1) and (6), identity (2) is equivalent to the following:

f_{λ}(a, b) = ∑_{μ∈H(λ)} b^{|λ/μ|}a^{c(μ)}.  (7)

Let A_{j}(λ) be the subdiagram of λ consisting of c_{j} columns of length j for j ≥ 1. Thus choosing a partition μ in H(λ) is equivalent to choose r leftmost (resp. the remaining c_{j} − r ) columns of length j (resp. j − 1 ) for μ within each block A_{j}(μ). Clearly the corresponding weight is

∑_{r=0}^{c_{j}} a^{c_{j}−r}b^{r} = \frac{a^{c_{j} +1}−b^{c_{j} +1}}{a−b}  if j is odd,
∑_{r=0}^{c_{j}}(ab)^{r} = \frac{1−(ab)^{c_{j} +1}}{1−ab}  if j is even.

Multiplying the weights on all j ≥ 1 yields (7).

Each pair (λ, μ) with μ ∈ H(λ) can be visualized by putting a cross (×) in each cell of λ/μ. Example. For λ = (10, 9, 8, 6, 1) and μ = (9, 8, 7, 3, 1) ∈ H(λ), their Ferrers diagrams and the block A_{4}(λ) are represented as follows :

Similarly, by (2) and (6), we see that theorem 5 is equivalent to the following:

f_{λ}(a, b, c) = ∑_{(μ, ν)∈C(λ)} a^{c(ν)}b^{|μ/ν|}c^{|λ/μ|},  (8)

[figure: λ =; A_{4}(λ) =]

where C(λ) = {(μ, ν) | μ ∈ H(λ), ν ∈ H(μ)}.

We shall compute the right-hand side of (8) using a sieve method. To this end we shall first enumerate a larger class of patterns whose generating function is equal to F_{Γ(λ)}(a, b, c).

Recall that we identify a partition λ with its Ferrers diagram. We will say that a subset S of ℕ^{2} is a partition diagram if {(x − k, y)|(x, y) ∈ S} is a Ferrers diagram for some integer k ≥ 0. Let H′(λ) be the set of all subsets μ of λ such that μ ∩ A_{j}(λ) is a partition diagram for all j ≥ 1 and λ/μ is a horizontal strip. Define

B(λ) = {(μ, ν) | μ ∈ H(λ), ν ∈ H′(μ)}.

Note that in the above definition, the subdiagram ν of λ is not necessarily a partition diagram. In this regard, the set C(λ) can be described as follows:

C(λ) = {(μ, ν) ∈ B(λ) | ν ∈ H(μ)}.

Given ν ∈ H′(μ), the jth row of ν is called compatible if

∀x ≥ 1, (x + 1; j) ∈ ν ⟹ (x; j) ∈ ν.

For p ≥ 0 let B_{p}(λ) be the set of (μ, ν) ∈ B(λ) such that ν has at least p non compatible rows. Clearly B_{0}(λ) = B(λ) and B(λ) \ C(λ) = B_{1}(λ), in other words, a pair (μ, ν) ∈ B(λ) is an element of C(λ) iff all the rows of ν are compatible. By the principle of inclusion-exclusion we obtain

∑_{(μ,ν)∈C(λ)} a^{c(ν)}b^{|μ/ν|}c^{|λ/μ|} = ∑_{p=0}^{l(λ)}(−1)^{p} ∑_{(μ,ν)∈B_{p}(λ)} a^{c(ν)}b^{|μ/ν|}c^{|λ/μ|}. (9)

Each triple (λ, μ, ν) with (μ, ν) ∈ B(λ) can be visualized by putting a circle ∘ (resp. cross ×) in each cell of μ/ν (resp. λ/μ). Example. The following diagrams represent two triples (λ, μ, ν) :

[figure: (a); (b)]

Clearly λ = (10, 9, 8, 6, 1) and μ = (9, 8, 7, 3). In (a), the pair (μ, ν) is in B_{1}(λ) because the third row of ν is not compatible, so ν ∈ H′(μ)\H(μ) and ν is not a partition. In (b), the pair (μ, ν) is in C(λ) because all the rows of ν are compatible, so ν = (8, 7, 7) is a partition in H(μ).

Lemma 1 We have

∑_{(μ,ν)∈B(λ)} a^{c(ν)}b^{|μ/ν|}c^{|λ/μ|} = F_{Γ(λ)}(a, b, c).

Proof. As in the proof of (7), we divide the diagram λ into rectangular blocks A_{j}(λ), j ≥ 1 , and compute the weight within each block A_{j}(λ). Clearly choosing a pair (μ, ν) in B(λ) is equivalent to, for each j ≥ 1, first choose the p left-most (resp. the remaining q = c_{j} − p) columns of length j (resp. j −1) for μ in A_{j}(λ), and then choose s (resp. p−s) left-most columns of length j (resp. j − 1) for ν among the p columns of μ, also choose r (resp. the remaining q − r) left-most columns of length j − 1 ( resp. j − 2) for ν. Thus the corresponding weight is h_{c_{1}}(a, b, c) if j = 1 and, for each j ≥ 2,

∑_{p+q=c_{j}} c^{q} (∑_{s=0}^{p}(ab)^{s}) (∑_{r=0}^{q} a^{r}b^{q−r}) = P_{c_{j}}(a, b, c) if j even;

∑_{p+q=c_{j}} c^{q} (∑_{s=0}^{p} b^{s}) (∑_{r=0}^{q}(ab)^{r}) = Q_{c_{j}}(a, b, c)  if j odd.

Multiplying up over all j ≥ 1 we get the desired formula. □

Example. Consider the (a) case of the previous example. The subdiagrams corresponding to the block A_{4}(λ) are the following :

[figure: A_{4}(λ) =; ⟶  {; μ ∩ A_{4}(λ) =; ν ∩ A_{4}(λ) =]

Note that c_{j} = 5, p = 2, s = 0 and r = 2.

For any set of integers J = {j_{1}, j_{2}, …, j_{p}} (p ≥ 1) let B_{J}(λ) denote the set of all the pairs (μ, ν) ∈ B(λ) such that the jth row of ν is not compatible for j ∈ J. Hence B_{J}(λ) ∈ B_{p}(λ).

Lemma 2 There holds

∑_{(μ,ν)∈B_{J} (λ)} a^{c(ν)}b^{|μ/ν|}c^{|λ/μ|} = (abc)^{p}δ_{j_{1}} … δ_{j_{p}}F_{Γ(λ)}(a, b, c).

Proof. Recall that λ′ = (1^{c_{1}}2^{c_{2}} …). Suppose there exists a pair (μ, ν) in B_{J}(λ), then there should be an integer x_{j} ∈ ℕ such that (x_{j} + 1, j) ∈ ν and (x_{j}, j) ∈ μ/ν for any j ∈ J. In view of the definition of B_{J}(λ) we must have x_{j} = c_{l(λ)} + ⋯ + c_{j+1} and (x_{j}, j + 1) ∈ λ/μ, for λ/μ is a horizontal strip. It follows that c_{j} ≥ 1 and c_{j+1} ≥ 1. Furthermore, if j + 1 is also in J, we must have c_{j+1} ≥ 2. Summarizing, we have the following equivalence:

B_{J}(λ) ≠ ∅ ⟺ c_{j}c_{j+1} ≠ 0 ∀j ∈ J and c_{j+1} ≥ 2 if j, j + 1 ∈ J.

It is easy to see that the last condition is equivalent to δ_{j_{1}} ⋯ δ_{j_{p}}Γ(λ) ∈ ℕ^{∞} or δ_{j_{1}} ⋯ δ_{j_{p}}F_{Γ(λ)}(a, b, c) ≠ 0.

In what follows we shall assume that B_{J}(λ) ≠ ∅. Thus we can define a unique partition δ_{J}(λ) such that Γ(δ_{J}(λ)) = δ_{j_{1}} … δ_{j_{p}}Γ(λ). Graphically, the diagram δ_{J}(λ) can be obtained by deleting, successively for j ∈ J, the x_{j}th and (x_{j} + 1)th columns and shift all the cells on the right of x_{j}th column of λ to left by two units. For (μ, ν) ∈ B_{J}(λ), if we apply the same graphical operation to the μ and ν, we get a pair (δ_{J}(μ), δ_{J}(ν)) ∈ B(δ_{J}(λ)). For example, in the previous example, if J = {3, 4}, then δ_{J}(λ) = (6, 5, 4, 3). The corresponding triples (λ, μ, ν) with (μ, ν) ∈ B(λ) and (δ_{J}(λ), δ_{J}(μ), δ_{J}(ν)) with (δ_{J}(μ), δ_{J}(ν)) ∈ B(δ_{J}(λ)) are illustrated as follows :

[figure: ⟶; \underbrace{deleted}; \underbrace{deleted}]

Since the weight corresponding to the deleted x_{j}th and x_{j+1}th columns of λ, μ and ν is abc for each j ∈ J, we have

a^{c(ν)}b^{|λ/ν|}c^{|μ/λ|} = (abc)^{p}a^{c(δ_{J} (ν))}b^{|δ_{J} (λ)/δ_{J} (ν|)}c^{|δ_{J} (μ)/δ_{J} (λ|)}.

Therefore

∑_{(λ,ν)∈B_{J} (λ)} a^{c(ν)}b^{|λ/ν|}c^{|μ/λ|} = (abc)^{p} ∑_{(λ,ν)∈B(δ_{J} (λ))} a^{c(ν)}b^{|λ/ν|}c^{|μ/λ|}.

The lemma follows then immediately from lemma 1. □

It follows from lemma 2 that for p ≥ 1

∑_{(μ,ν)∈B_{p}(λ)} a^{c(ν)}b^{|μ/ν|}c^{|λ/μ|} = (abc)^{p} ∑_{1≤j_{1}<···<j_{p}≤l(λ)} δ_{j_{1}} … δ_{j_{p}}F_{Γ(λ)}(a, b, c).

Combining with (9) and lemma 1 we derive immediately theorem 1.

Remark. Similarly, using another identity of Littlewood [7]:

∑_{λ} a^{r(λ)}s_{λ}(X) = ∏_{i} \frac{1 + ax_{i}}{1 − x_{i}^{2}} ∏_{j<k}(1 − x_{j}x_{k})^{−1},  (10)

where r(λ) is the number of rows of odd length of λ, we obtain:

∑_{λ} f_{λ′}(a, b) s_{λ}(X) = ∏_{i} \frac{(1 + ax_{i})(1 + bx_{i})}{1 − x_{i}^{2}} ∏_{j<k}(1 − x_{j}x_{k})^{−1}

and

∑_{λ} f_{λ′}(a, b, c) s_{λ}(X) = ∏_{i} \frac{(1 + ax_{i})(1 + bx_{i})(1 + cx_{i})}{1 − x_{i}^{2}} ∏_{j<k}(1 − x_{j}x_{k})^{−1}.

Note that c_{j}(λ′) = m_{j}(λ) is the multiplicity of j in λ.

3 Proof of theorem 5

Consider the generating function

S(u) = ∑_{λ_{0},λ} f_{λ}(a, b) s_{λ}(X) u^{λ_{0}}

where the sum is over all λ_{0} ≥ λ_{1} ≥ ⋯ ≥ λ_{n} ≥ 0, and λ = (λ_{1}, …, λ_{n}). Suppose λ is of form μ_{1}^{r_{1}}, μ_{2}^{r_{2}}, … μ_{k}^{r_{k}}, where μ_{1} > μ_{2} > ⋯ > μ_{k} ≥ 0 and the r_{i} are positive integers whose sum is n. Let S_{n}^{λ} = S_{r_{1}} × ⋯ S_{r_{k}} be the group of permutations leaving λ invariant. Then

s_{λ}(X) = ∑_{w∈S_{n}} w (x_{1}^{λ_{1}} … x_{n}^{λ_{n}} ∏_{i<j} \frac{x_{i}}{x_{i} − x_{j}} )

= ∑_{w∈S_{n}/S_{n}^{λ}} w (x_{1}^{λ_{1}} … x_{n}^{λ_{n}} ∏_{λ_{i}>λ_{j}} \frac{x_{i}}{x_{i} − x_{j}} ) ,

where the permutation w acts on the indices of the indeterminates. Each w ∈ S_{n}/S_{n}^{λ} corresponds to a surjective mapping f : X ⟶ {1, 2, …, k} such that |f^{−1}(i)| = r_{i}. For any subset Y of X, let p(Y) denote the product of the elements of Y. (In particular, p(∅) = 1.) We can rewrite Schur functions as follows:

s_{λ}(X) = ∑_{f} p(f^{−1}(1))^{μ_{1}} ⋯ p(f^{−1}(k))^{μ_{k}} ∏_{f(x_{i})<f(x_{j})} \frac{x_{i}}{x_{i} − x_{j}} ,

summed over all surjective mappings f : X ⟶ {1, 2, …, k} such that |f^{−1}(i)| = r_{i}. Furthermore, each such f determines a filtration of X:

F : ∅ = F_{0} ⊊ F_{1} ⊊ ⋯ ⊊ F_{k} = X,

according to the rule x_{i} ∈ F_{l} ⟺ f(x_{i}) ≤ l for 1 ≤ l ≤ k. Conversely, such a filtration F = (F_{0}, F_{1}, …, F_{k}) determines a surjection f : X ⟶ {1, 2, …, k} uniquely. Thus we can write:

s_{λ}(X) = ∑_{F} π_{F} ∏_{1≤i≤k} p(F_{i} \ F_{i−1})^{μ_{i}},  (11)

summed over all the filtrations F such that |F_{i}| = r_{1} + r_{2} + ⋯ + r_{i} for 1 ≤ i ≤ k, and

π_{F} = ∏_{f(x_{i})<f(x_{j})} \frac{x_{i}}{x_{i} − x_{j}} ,

where f is the function defined by F.

Now let ν_{i} = μ_{i} − μ_{i+1} if 1 ≤ i ≤ k − 1 and ν_{k} = μ_{k}, thus ν_{i} > 0 if i < k and ν_{k} ≥ 0. Since the lengths of columns of λ are |F_{j}| = r_{1} + ⋯ + r_{j} with multiplicities ν_{j} for 1 ≤ j ≤ k, we have

f_{λ}(a, b) = ∏_{|F_{j}| odd} \frac{a^{ν_{j}+1} − b^{ν_{j}+1}}{a − b} ∏_{|F_{j}| even} \frac{1 − (ab)^{ν_{j}+1}}{1 − ab} .  (12)

Furthermore, let μ_{0} = λ_{0} and ν_{0} = μ_{0} − μ_{1} in the definition of S(u), so that ν_{0} ≥ 0 and μ_{0} = ν_{0} + ν_{1} + ⋯ + ν_{k}. It follows from (11) and (12) that :

S(u) = ∑_{F} π_{F} ∑_{ν} u^{ν_{0}} ∏_{|F_{j}| odd} \frac{a^{v_{j}+1} − b^{v_{j}+1}}{a − b} u^{v_{j}}p(F_{j})^{v_{j}}

× ∏_{|F_{j}| even} \frac{1 − (ab)^{v_{j}+1}}{1 − ab} u^{v_{j}}p(F_{j})^{v_{j}},  (13)

where the outer sum is over all filtrations F of X and the inner sum is over all integers ν_{0}, ν_{1}, …, ν_{k} such that ν_{0} ≥ 0, ν_{k} ≥ 0 and ν_{i} > 0 for 1 ≤ i ≤ k − 1.

For any filtration F of X set

A_{F}(X, u) = ∏_{|F_{j}| odd} [\frac{a(a − b)^{−1}}{1 − ap(F_{j})u} − \frac{b(a − b)^{−1}}{1 − bp(F_{j})u} − χ(F_{j} ≠ X)]

× ∏_{|F_{j}| even≥1} [\frac{(1 − ab)^{−1}}{1 − p(F_{j})u} − \frac{ab(1 − ab)^{−1}}{1 − abp(F_{j})u} − χ(F_{j} ≠ X)] ,

where χ(A) = 1 if A is true, and χ(A) = 0 if A is false. Then the inner sum of (13) is

(1 − u)^{−1}A_{F}(X, u),

therefore

S(u) = (1 − u)^{−1} ∑_{F} π_{F}A_{F}(X, u),

where the sum is over all the filtrations of X as before.

The above formula shows that S(u) is a rational function of u whose denominator is the product of the form 1 − p(Y)u, 1 − ap(Y)u, 1 − bp(Y)u or 1 − abp(Y)u, where Y ⊆ X. Therefore we have the following result.

Lemma 3 The generating function S(u) is of the form:

S(u) = \frac{c(∅)}{1 − u} + ∑_{{Y ⊆X \atop |Y | odd}} (\frac{a(Y)}{1 − ap(Y)u} − \frac{b(Y)}{1 − bp(Y)u})

+ ∑_{{Y ⊆X \atop |Y | even>0}} (\frac{c(Y)}{1 − p(Y)u} − \frac{d(Y)}{1 − abp(Y)u}) .

It remains to compute the residues. Let us start with c(∅). Writing λ_{0} = λ_{1} + k with k ≥ 0, we see that

S(u) = ∑_{k≥0} u^{k} ∑_{λ} f_{λ}(a, b) s_{λ}(X)u^{λ_{1}}
= (1 − u)^{−1} ∑_{λ} f_{λ}(a, b) s_{λ}(X)u^{λ_{1}},

it follows from (2) that

c(∅) = (S(u)(1 − u)) |_{u=1} = Φ(X; a, b).

For computations of the other residues, we introduce some more notations. For any Y ⊆ X, let Y′ = X \ Y and −Y = {x_{i}^{−1} : x_{i} ∈ Y }. For any subset Z of X or −X let

α(Z, u) = \{ \begin{matrix}(1 − ap(Z)u)(1 − bp(Z)u) & if |Z| odd; \\ (1 − p(Z)u)(1 − abp(Z)u) & if |Z| even. \end{matrix}

As the computations of other residues are similar, we just give the details for c(Y). Let Y ⊆ X such that |Y | is even. Then we have

c(Y) = [(1 − u)^{−1} ∑_{F} π_{F}A_{F}(X; u)(1 − p(Y)u)]_{u=p(−Y)} .  (14)

If Y ∉ F, the corresponding summand is equal to 0. Thus we need only to consider the following filtrations F:

∅ = F_{0} ⊊ ⋯ ⊊ F_{t} = Y ⊊ ⋯ ⊊ F_{k} = X 1 ≤ t ≤ k.

We may then split F into two filtrations F_{1} and F_{2}, of −Y and Y′ = X \ Y respectively, as follows :

F_{1} : ∅ ⊊ −(Y \ F_{t−1}) ⊊ ⋯ ⊊ −(Y \ F_{1}) ⊊ −Y,
F_{2} : ∅ ⊊ F_{t+1} \ Y ⊊ ⋯ ⊊ F_{k−1} \ Y ⊊ Y′.

Then, writing v = p(Y)u, we have

(1 − u)^{−1}A_{F}(X; u)(1 − p(Y)u) = (1 − p(−Y)v)^{−1}A_{F_{1}}(−Y; v)A_{F_{2}}(Y′; v)

×α(−Y, v) [(1 − ab)^{−1} − β(v)(1 − v)] ,

where β(v) = ab/(1 − abv)(1 − ab) + χ(Y ≠ X), and

π_{F}(X) = π_{F_{1}}(−Y)π_{F_{2}}(Y′) ∏_{x_{i}∈Y,x_{j}∈Y′}(1 − x_{i}^{−1}x_{j})^{−1},

As u = p(−Y) is equivalent to v = 1, it follows from (14) that

c(Y) = (1 − ab)^{−1}(1 − p(−Y))^{−1}α(−Y, 1) ∏_{x_{i}∈Y,x_{j}∈Y′}(1 − x_{i}^{−1}x_{j})^{−1}

× [∑_{F_{1}} π_{F_{1}}(−Y)A_{F_{1}}(−Y; v)]_{v=1} × [∑_{F_{2}} π_{F_{2}}(Y′)A_{F_{2}}(Y′; v)]_{v=1} .

Using the result of c(∅), which can be written:

Φ(X; a, b) = ∑_{F} (π_{F}(X)A_{F}(X; u))_{u=1} ,

we obtain:

c(Y) = \frac{α(−Y, 1)Φ(−Y; a, b)Φ(Y′; a, b)}{(1 − ab)(1 − p(−Y))} ∏_{x_{i}∈Y,x_{j}∈Y′}(1 − x_{i}^{−1}x_{j})^{−1}.

Each subset Y of X can be encoded by a sequence ξ ∈ {±1}^{n} according to the rule : ξ_{i} = 1 if x_{i} ∉ Y and ξ_{i} = −1 if x_{i} ∈ Y. Hence

c(Y) = \frac{Φ(x_{1}^{ξ_{1}}, …, x_{n}^{ξ_{n}}; a, b)}{(1 − ab)(1 − p(−Y))}α(−Y, 1),

Note also that

p(Y) = ∏_{i} x_{i}^{(1−ξ_{i})/2}, p(−Y) = ∏_{i} x_{i}^{(ξ_{i}−1)/2}.

In the same way, we find for any even size subset Y ⊆ X that

d(Y) = \frac{abΦ(x_{1}^{ξ_{1}}, …, x_{n}^{ξ_{n}}; a, b)}{(1 − ab)(1 − (ab)^{−1}p(−Y))}α(−Y, 1),

and for any odd size subset Y ⊆ X that

a(Y) = \frac{aΦ(x_{1}^{ξ_{1}}, …, x_{n}^{ξ_{n}}; a, b)}{(a − b)(1 − a^{−1}p(−Y))}α(−Y, 1),
b(Y) = \frac{bΦ(x_{1}^{ξ_{1}}, …, x_{n}^{ξ_{n}}; a, b)}{(a − b)(1 − b^{−1}p(−Y))}α(−Y, 1).

By virtue of lemma 1, extracting the coefficient of u^{m} in S(u) yields

∑_{λ⊆(m^{n})} f_{λ}(a, b)s_{λ}(X) = Φ(X; a, b) + ∑_{{Y ⊆X \atop |Y | odd}} [a(Y)a^{m} − b(Y)b^{m}] p(Y)^{m}

+ ∑_{{Y ⊆X \atop |Y | even>0}} [c(Y) − d(Y)(ab)^{m}] p(Y)^{m}.

Finally, substituting the values of a(Y), b(Y), c(Y) and d(Y) in the above formula we obtain theorem 5.

4 Three special cases

First we note that f_{λ}(1, 0) = 1,

f_{λ}(1, −1) = \{ \begin{matrix}0 & if any c_{j} is odd, \\ 1 & otherwise; \end{matrix}

and

f_{λ}(0, 0) = \{ \begin{matrix}0 & if any c_{j} is positive for any odd j, \\ 1 & otherwise. \end{matrix}

On the other hand, we have

β(ξ, 1, 0) = 1,

β(ξ, 1, −1) = \{ \begin{matrix}1 & if m even, \\ ∏_{i} x_{i}^{(ξ_{i}−1)/2} & if m odd; \end{matrix}

and

β(ξ, 0, 0) = \{ \begin{matrix}0 & if |ξ|_{−1} is odd, \\ 1 & otherwise. \end{matrix}

So we derive immediately from theorem 5 the following result. Corollary 1 The sums of Schur functions of shape in a given rectangle are:

∑_{λ⊆(m)^{n}} s_{λ}(X) = ∑_{ξ∈{±1}^{n}} Φ(X^{ξ}; 1, 0) ∏_{i} x_{i}^{m(1−ξ_{i})/2},  (15)
∑_{{λ⊆(2m)^{n} \atop λ even}} s_{λ}(X) = ∑_{ξ∈\{±1\}^{n}} Φ(X^{ξ}; 1, −1) ∏_{i} x_{i}^{m(1−ξ_{i})},  (16)
∑_{{λ⊆(m)^{n} \atop λ′ even}} s_{λ}(X) = ∑_{{ξ∈\{±1\}^{n} \atop |ξ|_{−1} even}} Φ(X^{ξ}; 0, 0) ∏_{i} x_{i}^{m(1−ξ_{i})/2},  (17)

where n is even in the last identity.

To see that the above corollary is equivalent to theorems 2, 3 and 4, we need only to appeal to Vandermonde’s determinantal formula :

∑_{σ∈S_{n}} ϵ(σ) ∏_{i=1}^{n} x_{σ(i)}^{i−1} = det(x_{j}^{i−1}) = ∏_{1≤i<j≤n}(x_{j} − x_{i}).  (18)

Notice that for ξ ∈ {±1}^{n} and 1 ≤ i < j ≤ n, (x_{i}^{ξ_{i}} − x_{j}^{ξ_{j}})(x_{i}^{ξ_{i}}x_{j}^{ξ_{j}} − 1) = (x_{i} − x_{j})(x_{i}x_{j} − 1)x_{i}^{ξ_{i}−1}x_{j}^{ξ_{j}−1}, therefore

∏_{i<j}(x_{i}^{ξ_{i}} − x_{j}^{ξ_{j}})(x_{i}^{ξ_{i}}x_{j}^{ξ_{j}} − 1) = ∏_{i<j}(x_{i} − x_{j})(x_{i}x_{j} − 1) ∏_{i} x_{i}^{(n−1)(ξ_{i}−1)}. (19)

Note also that

∏_{i}(1 − x_{i}^{ξ_{j}}) = (−1)^{|ξ|_{−1}} ∏_{i<j}(1 − x_{i}) ∏_{i} x_{i}^{(ξ_{i}−1)/2}.  (20)

The (a, b) = (1, 0) case : Set

Δ_{B} = \frac{∏_{i<j}(x_{j} − x_{i})}{Φ(X; 1, 0)} = ∏_{i}(1 − x_{i}) ∏_{i<j}(x_{i} − x_{j})(x_{i}x_{j} − 1).

It follows from (18), (19) and (20) that

Φ(X^{ξ}; 1, 0) = \frac{(−1)^{|ξ|_{−1}}}{Δ_{B}} ∏_{i} x_{i}^{(1−ξ_{i})(n−1/2)} ∑_{σ∈S_{n}} ϵ(σ) ∏_{i} x_{σ(i)}^{ξ_{σ(i)}(i−1)}.

So the right side of (15) is

\frac{1}{Δ_{B}} ∑_{σ∈S_{n}} ϵ(σ) ∑_{ξ∈\{±1\}^{n}}(−1)^{|ξ|_{−1}} ∏_{i} x_{σ(i)}^{(m+2n−1)(1−ξ_{σ(i)})/2+ξ_{σ(i)}(i−1)}

= \frac{1}{Δ_{B}} ∑_{σ∈S_{n}} ϵ(σ) ∑_{ξ∈\{±1\}^{n}} ∏_{ξ_{σ(i)}=1} x_{σ(i)}^{i−1} ∏_{ξ_{σ(i)}=−1} (−x_{σ(i)}^{m+2n−i})

= \frac{1}{Δ_{B}} det (x_{i}^{j−1} − x_{i}^{m+2n−j}) .

Hence theorem 2 is equivalent to (15). The (a, b) = (1, −1) case : Set

Δ_{C} = \frac{∏_{i<j}(x_{j} − x_{i})}{Φ(X; 1, −1)} = ∏_{i}(1 − x_{i}^{2}) ∏_{i<j}(x_{i} − x_{j})(x_{i}x_{j} − 1).

It follows from (18), (19) and (20) that

Φ(X^{ξ}; 1, −1) = \frac{(−1)^{|ξ|_{−1}}}{Δ_{C}} ∏_{i} x_{i}^{n(1−ξ_{i})} ∑_{σ∈S_{n}} ϵ(σ) ∏_{i} x_{σ(i)}^{ξ_{σ(i)}(i−1)},

and the right hand side of (16) is

\frac{1}{Δ_{C}} ∑_{σ∈S_{n}} ϵ(σ) ∑_{ξ∈\{±1\}^{n}}(−1)^{|ξ|_{−1}} ∏_{i} x_{σ(i)}^{(n+m)(1−ξ_{σ(i)})+(i−1)ξ_{σ(i)}}

= \frac{1}{Δ_{C}} ∑_{σ∈S_{n}} ϵ(σ) ∑_{ξ∈\{±1\}^{n}} ∏_{ξ_{σ(i)}=1} x_{σ(i)}^{i−1} ∏_{ξ_{σ(i)}=−1} (−x_{σ(i)}^{2n+2m−i+1})

= \frac{1}{Δ_{C}} det (x_{i}^{j−1} − x_{i}^{2m+2n+1−j}) .

So Theorem 3 is equivalent to (16).

The (a, b) = (0, 0) case : Set

Δ_{D} = \frac{∏_{i<j}(x_{j} − x_{i})}{Φ(X; 0, 0)} = ∏_{i<j}(x_{i} − x_{j})(x_{i}x_{j} − 1).

It follows from (18), (19) and (20) that

Φ(X^{ξ}; 0, 0) = \frac{1}{Δ_{D}} ∏_{i} x_{i}^{(n−1)(1−ξ_{i})} ∑_{σ∈S_{n}} ϵ(σ) ∏_{i} x_{σ(i)}^{ξ_{σ(i)}(i−1)}

and the right side of (17) is

\frac{1}{Δ_{D}} ∑_{σ∈S_{n}} ϵ(σ) ∑_{{ξ∈\{±1\}^{n} \atop |ξ|_{−1}even}} ∏_{ξ_{σ(i)}=1} x_{σ(i)}^{i−1} ∏_{ξ_{σ(i)}=−1} x_{σ(i)}^{2n+m−i−1}

= \frac{1}{2Δ_{D}} [det (x_{i}^{j−1} − x_{i}^{m+2n−1−j}) + det (x_{i}^{j−1} + x_{i}^{m+2n−1−j})] .

So theorem 4 is equivalent to (17).

When m = 0, as the left sides of (15), (16) and (17) are equal to 1, we obtain the following result.

Corollary 2 For any non negative integer n, we have

det (x_{i}^{j−1} − x_{i}^{2n−j}) = ∏_{i}(1 − x_{i}) ∏_{i<j}(x_{i} − x_{j})(x_{i}x_{j} − 1),
det (x_{i}^{j−1} − x_{i}^{2n−j+1}) = ∏_{i}(1 − x_{i}^{2}) ∏_{i<j}(x_{i} − x_{j})(x_{i}x_{j} − 1),
det (x_{i}^{j−1} + x_{i}^{2n−1−j}) = 2 ∏_{i<j}(x_{i} − x_{j})(x_{i}x_{j} − 1).

These are actually Weyl’s denominator formulas for root systems of type B_{n}, C_{n} and D_{n} ([3], p. 68-69) respectively.

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