Littlewood’s formulas for characters of orthogonal and symplectic groups

Alain Lascoux

Institut Gaspard Monge, CNRS, Université de Marne-la-Vallée

77454 Marne-la-Vallée Cedex, France

e-mail: Alain.Lascoux @ univ-mlv.fr

Abstract

Litllewood gave expansions of products of the type ∏ 1/(1 − a_{i}a_{j}). Several authors published generalizations involving a small number of extra factors ∏ 1/(1 − xa_{i}). We show that the method of Littlewood (that we express in terms of λ-rings) covers these extensions. We also express Littlewood’s coefficients in terms of orthogonal or symplectic Schur functions.

Given an infinite alphabet 𝔸 = {a_{1}, a_{2}, a_{3}, …}, Littlewood [7] gave expansions of products of the type

∏(1 − a_{i})^{±1} ∏_{i<j}(1 − a_{i}a_{j})^{±1} & ∏(1 − a_{i})^{±1} ∏_{i≤j}(1 − a_{i}a_{j})^{±1} , (1)

the case ∏(1 − a_{i})^{−1} ∏_{i<j}(1 − a_{i}a_{j})^{−1} being due to Schur [10].

By functoriality of symmetric functions, Littlewood’s formulas imply without further computations expansions of more complicated products like

∏(1 − a_{i})^{±1} ∏(1 − b_{i})^{∓1} ∏_{i≤j}(1 − a_{i}a_{j})^{±1} ∏_{i<j}(1 − b_{i}b_{j})^{±1} ∏(1 − a_{i}b_{j})^{∓1} (2)

involving a second alphabet 𝔹 = {b_{1}, b_{2}, …}.

However, Littlewood’s method does not restrict to expliciting the above products, and allows to expand products of the type

∏_{i} \frac{1}{(1 − x_{1}a_{i}) ⋯ (1 − x_{n}a_{i})} ∏_{i<j} \frac{1}{1 − a_{i}a_{j}}  (3)

involving a finite alphabet 𝕏 = {x_{1}, …, x_{n}}, the alphabets 𝔸 and 𝕏 playing now a disymmetrical role.

To obtain his formulas, Littlewood describes in fact the adjoint of the multiplication by certain series of symmetric functions (see the theorem below). We shall state his results in terms of λ-rings.

The case 𝕏 = {x_{1}, x_{2}} has been considered by Ishikawa [2], Ishikawa and Wakayama [3], Bressoud [1], and the case 𝕏 = {x_{1}, x_{2}, x_{3}} by Jouhet and Zeng [4].

Before reinterpreting their formulas, let us recall some definitions, for which we refer to [8].

Given two alphabets 𝔸, 𝔹, the complete functions S_{j}(𝔸^{−}𝔹) are defined by the generating series (with z an extra variable)

σ_{z}(𝔸 − 𝔹) := ∏_{b∈𝔹}(1 − zb)/ ∏_{a∈𝔸}(1 − za)  (4)

Given λ = (λ_{1}, λ_{2}, …, λ_{r}) ∈ ℕ^{r}, the Schur function S_{λ}(𝔸^{−}𝔹) is

S_{λ}(𝔸^{−}𝔹) = |S_{λ_{i+j−i}}(𝔸^{−}𝔹)|_{1≤i,j≤r} .

When λ is weakly decreasing, it is called a partition. In that case, following Frobenius, one uses another description of partitions, by decomposing their diagram into its diagonal hooks, and writing λ = (α | β) [8] p.3.

On the space 𝔖ym(𝔸) of symmetric functions in 𝔸, there exists a scalar product ( , )_{𝔸}, and Schur functions indexed by partitions constitute an orthormal basis with respect to it :

( S_{λ}(𝔸), S_{λ}(𝔸))_{𝔸} = 1 & ( S_{λ}(𝔸), S_{ν}(𝔸))_{𝔸} = 0 λ ≠ ν . (5)

Another way to state (5) is to assert the existence of a reproducing kernel

K(𝔸, 𝔹) := σ_{1}(𝔸𝔹) = ∏_{a∈𝔸, b∈𝔹}(1 − ab)^{−1}

which satisfies

∀f ∈ 𝔖ym , ( K(𝔸, 𝔹), f(𝔸))_{𝔸} = f(𝔹) ,  (6)

because the Cauchy formula states that

K(𝔸, 𝔹) = ∑_{λ} S_{λ}(𝔸) S_{λ}(𝔹) ,  (7)

sum over all partitions.

Given any symmetric function f, the adjoint D(f; •) of the multiplication by f is defined by

∀f, g, h ∈ 𝔖ym , ( D(f; g) , h) = (g , fh)  (8)

The formulas

σ_{1}(Λ^{2}(𝔸)) = ∑ S_{μ}(𝔸) , all μ with columns of even lengths  (9)

and

σ_{1}(𝔸 + Λ^{2}(𝔸)) = ∑ S_{λ}(𝔸) , all partitions λ  (10)

are equivalent, because Λ^{2}(𝔸+1) = 𝔸+Λ^{2}(𝔸). Combinatorially, considering the letter 1 to be bigger than all the letters in 𝔸, it is equivalent to enumerate all tableaux in 𝔸, or all tableaux in the alphabet 𝔸 + 1 := 𝔸 ∪ {1} with columns of even length (we write unions of alphabets with a ‘+‘). Indeed, the 1’s occupy the horizontal strip such that the inner shape μ has columns of even lengths.

The computations of Littlewood in chapter XI of his book [7] can be rephrased in the λ-rings paradigm as follows :

Theorem 1 (Littlewood). Given two alphabets 𝔸, 𝕏, then one has the identities

σ_{1}(𝔸𝕏 + Λ^{2}(𝔸)) = ∑_{μ} S_{μ}(𝔸) D(σ_{1}(Λ^{2}); S_{μ}))(𝕏)  (11)
σ_{1}(𝔸𝕏 + S^{2}(𝔸)) = ∑_{μ} S_{μ}(𝔸) D(σ_{1}(S^{2}); S_{μ}))(𝕏)  (12)
σ_{1}(𝔸𝕏 − Λ^{2}(𝔸)) = ∑_{μ} S_{μ}(𝔸) D(σ_{1}(−Λ^{2}); S_{μ}))(𝕏)  (13)
σ_{1}(𝔸𝕏 − S^{2}(𝔸)) = ∑_{μ} S_{μ}(𝔸) D(σ_{1}(−S^{2}); S_{μ}))(𝕏) ,  (14)

sum over all partitions μ.

Proof. All four generating series factorize : σ_{1}(𝔸𝕏+Λ^{2}(𝔸)) = σ_{1}(𝔸𝕏)σ_{1}(Λ^{2}(𝔸)). Since σ_{1}(𝔸𝕏) is a reproducing kernel, one has for any μ :

( σ_{1}(𝔸𝕏 + Λ^{2}(𝔸)) , S_{μ}(𝔸))_{𝔸} = ( σ_{1}(𝔸𝕏) , D(σ_{1}(Λ^{2}); S_{μ})(𝔸))_{𝔸}
= D(σ_{1}(Λ^{2}); S_{μ})(𝕏) .

and this proves the first identity.

The same proof would work for σ_{1}(𝔸𝕏) f(𝔸), f being any symmetric function. Indeed property (6) entails

σ_{1}(𝔸𝕏) f(𝔸) = ∑_{μ} S_{μ}(𝔸) D(f; S_{μ})(𝕏) ,  (15)

and formulas (12, 13, 14) are a special case of it. QED

Let us mention that other expansions are possible. Indeed,

S^{2}(𝔸 ± 𝕏) = S^{2}(𝔸) ± S^{1}(𝔸)S^{1}(𝕏) + S^{2}(±𝕏)
Λ^{2}(𝔸 ± 𝕏) = Λ^{2}(𝔸) ± Λ^{1}(𝔸)Λ^{1}(𝕏) + Λ^{2}(±𝕏) .

It implies, for example, that

σ_{1}(𝔸𝕏 + Λ^{2}(𝔸)) = σ_{1}(−Λ^{2}(𝕏)) σ_{1}(Λ^{2}(𝔸 + 𝕏))

= σ_{1}(−Λ^{2}(𝕏)) ∑_{ν} S_{ν}(𝔸 + 𝕏) , (16)

sum over all partitions ν with columns of even lengths. To go further, one would decompose S_{ν}(𝔸 + 𝕏) in the basis of Schur functions in 𝔸, and use in general Littlewood-Richardson’s rule to collect functions in 𝕏. But in the case where 𝕏 is of small cardinality, the expansion can be written directly. For example ([1], th.IV), if 𝕏 = x_{1} + x_{2}, equation (16) can be rewritten as

σ_{1}((x_{1} + x_{2})𝔸 + Λ^{2}(𝔸)) = (1 − x_{1}x_{2}) ∑_{ν} S_{ν}(𝔸 + x_{1} + x_{2})  (17)

and is easy to expand in terms of the S_{μ}(𝔸), interpreting it as the sum of all Young tableaux with even columns, on an alphabet with last two letters x_{1}, x_{2}.

Let us show in more details how to make explicit the coefficients in Littlewood’s identities, for card(𝕏) = n = 1, 2, 3, recovering the related results of [1, 2, 3, 4]. We shall treat only the case of σ_{1}(𝕏𝔸 + Λ^{2}𝔸), the other being similar.

When n = 1, 𝕏 = {x_{1}}. Then D(S_{ν}; S_{μ})(𝕏) is non zero iff the diagram of μ/ν is an horizontal strip. Therefore, given any partition μ, there is one and only one partition ν with even columns such that D(S_{ν}; S_{μ})(𝕏) is non zero. Equation (11) specializes to

σ_{1}(x_{1}𝔸 + Λ^{2}(𝔸)) = ∑_{μ} x_{1}^{odd(μ)} S_{μ}(𝔸) ,  (18)

where odd(μ) is the number of columns of μ of odd length.

Take now n = 2. A skew diagram μ/ν gives a non-zero function S_{μ/ν}(x_{1} ^{+}x_{2}) iff it is made of horizontal strips of height 1 and 2. Decompose the partition into blocks B_{k}, k = 1, 2, … of columns of the same length k (denoting ℓ_{k} its width). If k is odd, then the restriction of μ/ν to this block must be an horizontal strip of length ℓ_{k}, giving a factor S_{ℓ_{k}}(x_{1} + x_{2}). If k is even, the restriction of μ/ν is a 2 × i rectangle, with 0 ≤ i ≤ ℓ_{k}. The figure shows the restriction of a strip to two consecutive blocks :

^{i}

□ ··· □ ■ ··· ■
□ ··· □ ■ ··· ■ ■ ··· ■
□ ··· □ □ ··· □ □ ··· □
⋮ ⋮ ⋮ ⋮ ⋮ ⋮ □ ··· □ □ ··· □ □ ··· □ _{ℓ_{2r}} _{ℓ_{2r−1}}

The sum over all possible i gives a factor S_{ℓ_{k}}(1 + x_{1}x_{2}). Moreover, there is no interference between the different blocks, and therefore, one gets that the coefficient of S_{μ}(𝔸) in σ_{1}((x_{1} ^{+}x_{2})𝔸 + Λ^{2}(𝔸)) is

D(σ_{1}(Λ^{2}); S_{μ})(x_{1} + x_{2}) = ∏_{k=0}^{∞} S_{ℓ_{2k}}(1 + x_{1}x_{2}) S_{ℓ_{2k+1}}(x_{1} + x_{2}) .  (19)

The case n = 3 is treated by Jouhet and Zeng [4]. The independent blocks are now the blocks B_{k} of columns of length 2k ^{+}1, 2k.

There are two possible configurations for those μ/ν giving a non-zero contribution :

^{i} ; \begin{matrix}■ & ··· & ■ & ■ & ··· & ■ \\ □ & ··· & □ & ■ & ··· & ■ & ■ & ··· & ■ \\ □ & ··· & □ & ■ & ··· & ■ & ■ & ··· & ■ \\ □ & ··· & □ & □ & ··· & □ & □ & ··· & □ \\ ⋮ & & ⋮ & ⋮ & & ⋮ & ⋮ & & ⋮ \\ □ & ··· & □ & □ & ··· & □ & □ & ··· & □ \end{matrix} ; _{_{p} _{q}} , p = ℓ_{2k+1}, q = ℓ_{2k}, i ≥ 1

The sum over all i = 1, …, p gives

x_{1}x_{2}x_{3}S_{p−1}(1 + x_{1}x_{2}x_{3})S_{q}(Λ^{2}(x_{1} + x_{2} + x_{3})) .

^{j} ; \begin{matrix}■ & ■ & ··· & ■ \\ □ & □ & ··· & □ & □ & ··· & □ & ■ & ··· & ■ \\ □ & □ & ··· & □ & □ & ··· & □ & ■ & ··· & ■ \\ □ & □ & ··· & □ & □ & ··· & □ & □ & ··· & □ \\ ⋮ & ⋮ & & ⋮ & ⋮ & & ⋮ & ⋮ & & ⋮ \\ □ & □ & ··· & □ & □ & ··· & □ & □ & ··· & □ \end{matrix} ; _{_{p} _{q}} , i = 0, 0 ≤ j ≤ q .

The sum over all j = 0, …, q is

S_{p}(x_{1} + x_{2} + x_{3}) S_{q}(1 + Λ^{2}(𝕏)) .

For example, for μ = [7, 7, 4], the possible configurations, together with their factors, are :

S_{411}(𝕏) S_{33}(𝕏)^{\begin{matrix}■ & ■ & ■ & ■ \\ □ & □ & □ & ■ & ■ & ■ & ■ \\ □ & □ & □ & ■ & ■ & ■ & ■ \end{matrix}} + S_{422}(𝕏) S_{33}(𝕏)^{\begin{matrix}■ & ■ & ■ & ■ \\ □ & □ & ■ & ■ & ■ & ■ & ■ \\ □ & □ & ■ & ■ & ■ & ■ & ■ \end{matrix}} + S_{433}(𝕏) S_{33}(𝕏)^{\begin{matrix}■ & ■ & ■ & ■ \\ □ & ■ & ■ & ■ & ■ & ■ & ■ \\ □ & ■ & ■ & ■ & ■ & ■ & ■ \end{matrix}} + S_{444}(𝕏) S_{33}(𝕏)^{\begin{matrix}■ & ■ & ■ & ■ \\ ■ & ■ & ■ & ■ & ■ & ■ & ■ \\ ■ & ■ & ■ & ■ & ■ & ■ & ■ \end{matrix}}

+ S_{4}(𝕏) S_{00}(𝕏)^{\begin{matrix}■ & ■ & ■ & ■ \\ □ & □ & □ & □ & □ & □ & □ \\ □ & □ & □ & □ & □ & □ & □ \end{matrix}} + S_{4}(𝕏) S_{11}(𝕏)^{\begin{matrix}■ & ■ & ■ & ■ \\ □ & □ & □ & □ & □ & □ & ■ \\ □ & □ & □ & □ & □ & □ & ■ \end{matrix}} + S_{4}(𝕏) S_{22}(𝕏)^{\begin{matrix}■ & ■ & ■ & ■ \\ □ & □ & □ & □ & □ & ■ & ■ \\ □ & □ & □ & □ & □ & ■ & ■ \end{matrix}} + S_{4}(𝕏) S_{33}(𝕏)^{\begin{matrix}■ & ■ & ■ & ■ \\ □ & □ & □ & □ & ■ & ■ & ■ \\ □ & □ & □ & □ & ■ & ■ & ■ \end{matrix}} .

In summary, the contribution of the block B_{k} is

c(B_{k}) = ∑_{i=0}^{p} S_{p−i}(𝕏)S_{qq}(𝕏) + S_{p}𝕏)(1 + S_{11}(𝕏) + ⋯ + S_{nn}(𝕏)) (20)

and the coefficient of S_{μ}(𝔸) in σ_{1}((x_{1} ^{+}x_{2} ^{+}x_{3})𝔸+Λ^{2}(𝔸)) is the product of all the factors c(B_{k}) given by the decomposition of μ into blocks. This expression is equivalent to formula 8 of Jouhet and Zeng.

Littlewood’s expansions were motivated by the description of characters of orthogonal and symplectic groups. In turn, these characters allow to expand Littlewood’s formulas (13), (14).

Following [7], p.240, using a second alphabet 𝕏, define the orthogonal Schur function O_{λ}(𝔸) by the generating series :

∏_{i≤j}(1 − a_{i}a_{j}) ∏_{i} ∏_{j}(1 − a_{i}x_{j})^{−1} = ∑_{λ} O_{λ}(𝕏) S_{λ}(𝔸) ,  (21)

that is,

σ_{1}(𝔸𝕏 − S^{2}(𝔸)) = ∑_{λ} O_{λ}(𝔸) S_{λ}(𝕏) .  (22)

Formula (14) can now be rewritten :

O_{λ} = D(σ_{1}(−S^{2}) ; S_{λ})  (23)

and the inverse formula is

S_{λ} = subs(S = O, D(σ_{1}(S^{2}) ; S_{λ}))  (24)

which means that in that last case, one uses the operator D(σ_{1}(S^{2}) ; •) as a formal operator on a vector space, the basis of which is indexed by partitions (and the action being obtained by identifying this basis with Schur functions).

For example, Littlewood (p. 241) gives (writing in the expansion of σ_{1}(−S^{2}) and σ_{1}(S^{2}) only the terms which give a non-zero contribution) :

O_{332} = D(S_{0}^{−}S_{2} ^{+}S_{31}^{−}S_{33} ; S_{332}) = S_{332} − S_{332/2} + S_{332/31} − S_{2}

and the identity

D(S_{0} ^{+}S_{2} ^{+}S_{22} ^{+}S_{222} ; S_{332}) = S_{332} + S_{33} + S_{321} + S_{211} + S_{31} + S_{11}

implies, by formal change of S into O :

S_{332} = O_{332} + O_{33} + S_{321} + O_{211} + O_{31} + O_{11} .

Ishikawa and Wakayama [3] give the expansion of σ_{1}((x_{1} +x_{2})𝔸−S^{2}(𝔸)), with x_{1}x_{2} = 1. It means computing the specialization of orthogonal Schur functions for such 𝕏. In fact the complete functions (resp. power sums) of 𝕏 are Tchebychef polynomials of the first and second kind in the variable (x_{1} + x_{2})/2. Ishikawa and Wakayama characterize those partitions for which O_{λ}(𝕏) ≠ 0 in terms of their Frobenius decomposition, and obtains that in that case ±O_{λ}(𝕏) is equal to a Tchebychef polynomial of the second kind.

Using symmetrizing operators, one easily expands σ_{1}(−S^{2}) as :

σ_{1}(−S^{2}) = ∑ ±S_{ϵ_{1},ϵ_{2},ϵ_{3},...} = S_{000...} − S_{200...} − S_{040...} + S_{240...} + ⋯ (25)
= S_{0} − S_{2} + S_{31} − S_{33} + · · · .  (26)

In the first summation, Schur functions are indexed by vectors [ϵ_{1}, ϵ_{2}, ϵ_{3}, …] such that ϵ_{i} ∈ {0, 2i}, and not by partitions, and the sign alternates with the parity of the number of non-zero components.

It is remarkable that the operator D(σ_{1}(−S^{2}); •) can also be written as an enumeration of vectors :

S_{λ} → O_{λ} = D(σ_{1}(−S^{2}), S_{λ}) = ∑_{ϵ_{1},ϵ_{2},...} ±S_{λ/[ϵ_{1},ϵ_{2},...]}  (27)

(recall that, for any two integral vectors λ, ν, S_{λ/ν} stands for the determinant |S_{λ_{i}+j−i−ν_{j}} |).

Indeed, the fact that D(S_{ν}; S_{λ}) = S_{λ/ν} for two partitions, extends, by reordering the rows of the determinantal expression of S_{ν} and the columns of S_{λ/ν}, to the case where ν is not a partition, i.e. is not decreasing.

Thus O_{332} can also be written

O_{332} = S_{332/000} − S_{332/200} − S_{332/040} + S_{332/240}

= | \begin{matrix}S_{3} & S_{4} & S_{5} \\ S_{2} & S_{3} & S_{4} \\ S_{0} & S_{1} & S_{2} \end{matrix} | − | \begin{matrix}S_{1} & S_{4} & S_{5} \\ S_{0} & S_{3} & S_{4} \\ 0 & S_{1} & S_{2} \end{matrix} | − | \begin{matrix}S_{3} & S_{0} & S_{5} \\ S_{2} & 0 & S_{4} \\ S_{0} & 0 & S_{2} \end{matrix} | + | \begin{matrix}S_{1} & S_{0} & S_{5} \\ S_{0} & 0 & S_{4} \\ 0 & 0 & S_{2} \end{matrix} | .

By mutilinearity of determinants, one can write the above sum as a single determinant, due to Weyl ([11], th.7.9.A), that we display for the order 3 :

O_{λ_{1}λ_{2}λ_{3}} = | \begin{matrix}S_{λ_{1}} − S_{λ_{1}−2} & S_{λ_{1}+1} − S_{λ_{1}+1−4} & S_{λ_{1}+2} − S_{λ_{1}+2−6} \\ S_{λ_{2}−1} − S_{λ_{2}−1−2} & S_{λ_{2}} − S_{λ_{2}−4} & S_{λ_{2}+1} − S_{λ_{2}+1−6} \\ S_{λ_{3}−2} − S_{λ_{3}−2} & S_{λ_{3}−1} − S_{λ_{3}−1−4} & S_{λ_{3}} − S_{λ_{3}−6} \end{matrix} | (28)

For example, O_{332}, again, is equal to

|\begin{matrix}S_{3} − S_{1} & S_{4} − S_{0} & S_{5} − 0 \\ S_{2} − S_{0} & S_{3} − 0 & S_{4} − 0 \\ S_{0} − 0 & S_{1} − 0 & S_{2} − 0 \end{matrix}|

We shall refer to [5] for determinantal expressions of orthogonal or symplectic characters. The case of “nearly rectangular shape” leads to an interesting combinatorics obtained by Krattenthaler [6].

Similarly, define the symplectic Schur function Sp_{λ} by

Sp_{λ} = D(σ_{1}(−Λ^{2}) ; S_{λ}) ,  (29)

but this time, it will be evaluated into the alphabet 𝕐 := {x_{1}, 1/x_{1}, x_{2}, 1/x_{2}, …, x_{n}, 1/x_{n}} instead of 𝕏.

The identity

σ_{1}(−Λ^{2}) = ∑ S_{ϵ_{0},ϵ_{1},ϵ_{2},ϵ_{3},...} ,  (30)

sum over all ϵ_{i} ∈ {0, 2i}, with ϵ_{0} = 0, extends, as before, to

Sp_{λ} = D(σ_{1}(−Λ^{2}) ; S_{λ}) = ∑_{ϵ_{1},ϵ_{2},...} S_{λ/[0,ϵ_{1},ϵ_{2},...]} ,  (31)

and this sum can be written as a single determinant due to Weyl ([11], th.7.8.E) :

|\begin{matrix}S_{λ_{1}} & S_{λ_{1}+1} − S_{λ_{1}+1−2} & S_{λ_{1}+2} − S_{λ_{1}+2−4} \\ S_{λ_{2}−1} & S_{λ_{2}} − S_{λ_{2}−2} & S_{λ_{2}+1} − S_{λ_{2}+1−4} \\ S_{λ_{3}−2} & S_{λ_{3}−1} − S_{λ_{3}−1−2} & S_{λ_{3}} − S_{λ_{3}−4} \end{matrix} |  (32)

Lovers of λ-rings will prefer the expression

Sp_{λ} = S_{λ}(𝕐 − 0, 𝕐 − 2, 𝕐 − 4, …) := |S_{λ_{i}+j−i}(𝕐 − 2j + 2)| ,  (33)

with, for any alphabet 𝕐, any number k,

σ_{z}(𝕐 − k) := σ_{z}(𝕐 − 1 − ⋯ − 1) = (1 − z)^{k} σ_{z}(𝕐) .

Indeed, because

S_{i}(𝕐 − k) = S_{i}(𝕐) − \binom{k}{1}S_{i−1}(𝕐) + \binom{k}{2}S_{i−2}(𝕐) − ⋯ ± S_{i−k}(𝕐) ,

expanding each term of the determinant |S_{λ_{i}+j−i}(𝕐 − 2j + 2)|, one obtains Weyl’s determinant by eliminating in each column multiples of the columns on its left.

In particular, when x_{1}, …, x_{n} are specialized to 1, the determinant specializes to a determinant of binomial coefficients

| \binom{2n + λ_{i} − i − j + 1}{λ_{i} + j − i} |_{1≤i,j≤n}  (34)

which is one of the many expressions of the dimension of a representation of 𝔖p_{2n}.

Let us mention that the interpretation of the orthogonal and symplectic Schur functions as characters depends on the parity of the number of variables. See Proctor [9] for the “‘odd symplectic groups”.

References

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[2] M. Ishikawa Minor summation formula of Pfaffians and Schur functions identities, FPSAC 95 (Formal Power Series and Algebraic Combinatorics), Université de Marne La Vallée (1995) 379–386.

[3] M. Ishikawa, M. Wakayama. Applications of Minor-Summation Formula II. Pfaffians and Schur Polynomials, J. Combin. Th., Ser. A 88 (1999) 136-157.

[4] F. Jouhet, J. Zeng. Some new identities for Schur functions preprint (2001).

[5] K. Koike, I. Terada. Young diagrammatic methods for the representation theory of the classical groups of type B_{n}, C_{n}, D_{n}, J. Algebra 107 (1987) 466–511.

[6] C. Krattenthaler. Identities for classical group characters of nearly rectangular shape, J. Algebra 209 (1998) 1–64.

[7] D.E. Littlewood. The theory of group characters, Oxford University Press (1950).

[8] I.G. Macdonald. Symmetric functions and Hall polynomials, Clarendon Press, second edition, Oxford, (1995).

[9] R. Proctor. Interconnections between symplectic and orthogonal characters, Contemp. Math. 88 (1989) 145–162.

[10] I. Schur. Gesammelte Abhandlungen, vol III, p.456, Springer (1973).

[11] H. Weyl. The classical groups, Princeton Univ. Press (1939).