POWERS OF STAIRCASE SCHUR FUNCTIONS AND SYMMETRIC ANALOGUES OF BESSEL POLYNOMIALS

Bernard LECLERC †

Abstract. We present several identities involving staircase Schur functions. These identities are then interpreted in terms of a sequence of orthogonal polynomials in one variable x, with coefficients in the ring of symmetric functions. By an appropriate specialization these polynomials reduce to Bessel polynomials. This leads to a new determinantal expression for Bessel polynomials and suggests that their combinatorics might be linked to Young tableaux or shifted Young tableaux.

1 Introduction

Schur functions S_{I} and skew Schur functions S_{I/J} are indexed by partitions I or skew partitions I/J, which are visualized graphically by a diagram of boxes. Several families of Schur functions associated with special diagrams are known to satisfy particular identities. Thus, Lascoux and Pragacz provided a determinantal expression of Schur functions and skew Schur functions in terms of ribbon functions, which generalizes the classical decomposition into hook functions given by Giambelli [LP].

In this paper we are interested in Schur functions whose diagram is a staircase. Our starting point is the following formula for the square of a staircase Schur function

S_{12...n}^{2} = S_{12...n+1/1}S_{12...n−1} − S_{12...n+1}S_{12...n−1/1} ,

which first arose in the context of the Korteweg-de-Vries equation [AM]. It shows that the polynomials θ_{d} constructed by Adler and Moser to represent the rational solutions of the KdV equation are, up to normalization, nothing but staircase Schur functions *. In a recent article Goulden has provided a combinatorial proof of this identity [Go]. We shall present more general formulae, expressing any power of a staircase Schur function.

An interpretation of these formulae in terms of orthogonal polynomials is then given. Indeed, it is shown that the family of staircase Schur functions S_{12...n}(𝔼 + x), n ≥ 0, is a family of orthogonal polynomials in the variable x, which may be seen as a symmetric analogue of the family of Bessel polynomials. This provides a new determinantal expression for Bessel polynomials. It also shows that the powerful machinery of symmetric functions might be of some help in the study of this sequence of polynomials. Thus, using the fact that staircase S-functions are special cases of Schur Q-functions, Thibon and the author derived from a recent multiplication formula due to Worley the computation of the linearization coefficients [LT]which had been conjectured by Favreau [Fa].

† Institut Gaspard Monge, Université de Marne-la-Vallée, 2 rue de la Butte-Verte 93160 Noisy-le-Grand Cedex France * We thank D. Svrtan for pointing out this fact to us and kindly providing references on this subject.

Finally, we show that our formulae are also linked to an old problem first studied by Borchardt [Bo] and Laguerre [La], and further investigated by numerous authors including Pólya [Po] and Foulkes [Fo]. The problem is to express any symmetric polynomial of n indeterminates as a rational function of the power sums of odd degree. Set 𝕏 = {x_{1}, …,x_{n}}. The expression given by Polya and Foulkes for the k^{th} elementary symmetric polynomial Λ_{k}(𝕏) is the coefficient of (−x)^{n−k} in the ratio S_{12...n}(𝕏 − x)/S_{12...n−1}(𝕏). We obtain a similar expression for the complete symmetric polynomial S_{k}(𝕏).

2 Schur functions and minors identities

In this section we shall define our notations and recall some basic facts about Schur functions and minors identities. General references are [LS], [Mc] and [Le].

Let 𝔼 denote a set of indeterminates or alphabet. The complete symmetric functions S_{i}(𝔼) and the elementary symmetric functions Λ_{i}(𝔼) are defined by means of the generating series

∑_{i} S_{i}(𝔼)t^{i} = ∏_{e∈𝔼}(1 − te)^{−1} ,  (1)

∑_{i} Λ_{i}(𝔼)t^{i} = ∏_{e∈𝔼}(1 + te) .  (2)

When there is no danger of confusion we shall omit the mention of the alphabet. For I = (i_{1}, …,i_{n}) ∈ ℕ^{n}, the Schur function S_{I} is defined by Jacobi–Trudi identity S_{I} = det[S_{i_{l}+l−k}]_{1≤k,l≤n} . In other words, denoting by S the infinite Toeplitz matrix [S_{j−i}]_{i,j≥0} where S_{k} = 0 for k < 0, S_{I} is the minor of S taken on the lines 0, 1, …, n − 1 and the columns i_{1}, i_{2} +1, …, i_{n} +n−1. More generally, given J = (j_{1}, …, j_{n}) ∈ ℕ^{n} one defines the skew Schur function S_{I/J} as the minor of S taken on the lines j_{1}, j_{2} + 1, …, j_{n} + n − 1 and the columns i_{1}, i_{2} + 1, …, i_{n} + n − 1 , i. e. S_{I/J} = det[S_{i_{l}+l−j_{k}−k}]_{1≤k,l≤n} . Note that this definition makes sense even if the sequences I and J are not arranged in increasing order. However, by permutation of columns and rows one can always assume that I and J are partitions i.e. weakly increasing sequences of nonnegative integers. A partition I is represented by a diagram of boxes having i_{1} boxes on its first row, i_{2} boxes on its second row, etc.. |I| = i_{1} + … + i_{n} is called the weight of I. The skew diagram associated with S_{I/J} is the complement of the diagram J in the diagram I.

Schur functions may also be expressed in terms of elementary symmetric functions Λ_{i}. Indeed, defining by analogy the functions Λ_{I/J} by Λ_{I/J} = det[Λ_{i_{l}+l−j_{k}−k}]_{1≤k,l≤n} , one has the relation

Λ_{I/J} = S_{I^{∼}/J^{∼}},  (3)

where I^{∼} denotes the conjugate partition of I i.e. the partition whose diagram is obtained by interchanging the rows and columns of the diagram of I.

Suppose 𝔽 is a second alphabet and denote by 𝔼 + 𝔽 the union of 𝔼 and 𝔽. It follows from (1), (2) that

S_{n}(𝔼 + 𝔽) = ∑_{0≤i≤n} S_{i}(𝔼)S_{n−i}(𝔽) ,  (4)

Λ_{n}(𝔼 + 𝔽) = ∑_{0≤i≤n} Λ_{i}(𝔼)Λ_{n−i}(𝔽) .  (5)

We shall also define the difference 𝔼 − 𝔽 by setting

∑_{i} S_{i}(𝔼 − 𝔽)t^{i} = \frac{∏_{f∈𝔽}(1 − tf)}{∏_{e∈𝔼}(1 − te)} ,  ∑_{i} Λ_{i}(𝔼 − 𝔽)t^{i} = \frac{∏_{e∈𝔼}(1 + te)}{∏_{f∈𝔽}(1 + tf)} ,

so that in particular

Λ_{i}(𝔼) = (−1)^{i}S_{i}(−𝔼) ,  (6)

which is another way of expressing the duality between complete and elementary symmetric functions. This yields a compact expression for the polynomial whose set of zeros is the alphabet 𝔸 of cardinality n

∏_{a∈𝔸}(x − a) = ∑_{i} x^{n−i}(−1)^{i}Λ_{i}(𝔸) = ∑_{i} S_{n−i}(x)S_{i}(−𝔸) = S_{n}(x − 𝔸) = (−1)^{n}Λ_{n}(𝔸 − x) .

Taking different alphabets 𝔼_{1}, …, 𝔼_{n} in the columns of the determinant of Jacobi–Trudi formula, one gets a multi-Schur function S_{I}(𝔼_{1}, …, 𝔼_{n}) = det[S_{i_{l}+l−k}(𝔼_{l})]_{1≤k,l≤n} . An example of multi-Schur function considered in this paper is the n^{th} orthogonal polynomial associated with the linear functional μ of moments μ(x^{i}) = S_{i}(𝔼). (Note that, since the S_{i} are algebraically independent, one may always define a formal alphabet 𝔼 by assigning to each S_{i} a given value S_{i}(𝔼) in a commutative ring.) Indeed this polynomial p_{n}(x) has, up to a constant factor, the expression

p_{n}(x) = S_{n^{n}0}(𝔼,𝔼, …,𝔼,x) = | \begin{matrix}S_{n}(𝔼) & S_{n+1}(𝔼) & … & S_{2n−1}(𝔼) & x^{n} \\ S_{n−1}(𝔼) & S_{n}(𝔼) & … & S_{2n−2}(𝔼) & x^{n−1} \\ ⋮ & ⋮ & & ⋮ & ⋮ \\ S_{0}(𝔼) & S_{1}(𝔼) & … & S_{n−1}(𝔼) & 1 \end{matrix} | , (7)

(see [Sz] p.27). By subtraction of lines in the determinant, this polynomial may also be written as an ordinary Schur function on the alphabet 𝔼 − x. Indeed, one has p_{n}(x) = S_{n^{n}}(𝔼 − x) . This is a consequence of the following important lemma.

Lemma 2.1 Let m, n be two integers m ≤ n, 𝔽 an alphabet of cardinality m, and I = (i_{1}, …, i_{n}) a partition. Then

S_{I}(𝔼_{1}, …,𝔼_{n}) = | \begin{matrix}S_{i_{1}}(𝔼_{1} − 𝔽) & … & S_{i_{n}+n−1}(𝔼_{n} − 𝔽) \\ ⋮ & & ⋮ \\ S_{i_{1}−n+m+1}(𝔼_{1} − 𝔽) & … & S_{i_{n}+m}(𝔼_{n} − 𝔽) \\ S_{i_{1}−n+m}(𝔼_{1}) & … & S_{i_{n}+m−1}(𝔼_{n}) \\ ⋮ & & ⋮ \\ S_{i_{1}−n+1}(𝔼_{1}) & … & S_{i_{n}}(𝔼_{n}) \end{matrix} | .

Proof. By (4), we have S_{k}(𝔼_{j} − 𝔽) = S_{k}(𝔼_{j}) + S_{k−1}(𝔼_{j})S_{1}(−𝔽) + … + S_{k}(−𝔽) . But S_{k}(−𝔽) = (−1)^{k}Λ_{k}(𝔽) = 0 if k > m. Thus, the transformation consists in adding to each of the first n − m rows a linear combination of the next m rows.

The Jacobi–Trudi formula expresses Schur functions as minors of the matrix S of complete symmetric functions S_{i}. This shows that many algebraic relations for Schur functions may be obtained by merely specializing the numerous identities satisfied by minors of a generic matrix. We have shown elsewhere [Le] that a great number of these identities are easily derived from a single one by Turnbull. We shall briefly recall this identity for the convenience of the reader. We first need some appropriate notations.

Let M be a n × ∞ matrix. We shall be interested in identities satisfied by maximal minors of M. Let a,b, …,c be n column vectors of M. The maximal minor of M taken on these n columns will be denoted by either a bracket or a one–line tableau

[ab … c] = a b … c .

A product of p minors of M will be designated by a p × n tableau

[ab … c].[de … f].[gh … i] = \begin{matrix}a & b & … & c \\ d & e & … & f \\ g & h & … & i \end{matrix} .

To denote certain alternating sums of products of minors, we shall use tableaux with boxes enclosing some particular vectors. They are defined in the following way.

Let T be a p × n tableau and A a subset of elements of T. Given a permutation σ of the elements of A, denote by σ(T) the tableau in which the elements of A are permuted by σ. Now, boxing in T the elements of A, we get a new tableau τ which will represent the alternating sum of all the tableaux σ(T), taking into account the fact that a permutation of elements of the same row gives a trivial action. More precisely, τ = ∑_{σ} sign(σ).σ(T) , where σ runs through the cosets of the symmetric group 𝔖(A) modulo the subgroup of the permutations which leave unchanged the rows of T.

Let i_{k}, k = 1, … p denote the number of elements of A lying in the k^{th} row of τ. The number of terms in this sum will be (i_{1} + … + i_{p})!/i_{1}! … i_{p}!. In particular, if all the enclosed elements of τ lie in the same row, it reduces to the single tableau T.

To illustrate these notations, let us write a well–known minor identity (Plücker’s relations) and then its transcription with tableaux.

Let M be the matrix

M = ( \begin{matrix}a_{1} & b_{1} & c_{1} & d_{1} & e_{1} & f_{1} & … \\ a_{2} & b_{2} & c_{2} & d_{2} & e_{2} & f_{2} & … \\ a_{3} & b_{3} & c_{3} & d_{3} & e_{3} & f_{3} & … \end{matrix} ) .

We have

| \begin{matrix}a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{matrix} | . | \begin{matrix}d_{1} & e_{1} & f_{1} \\ d_{2} & e_{2} & f_{2} \\ d_{3} & e_{3} & f_{3} \end{matrix} | − | \begin{matrix}e_{1} & b_{1} & c_{1} \\ e_{2} & b_{2} & c_{2} \\ e_{3} & b_{3} & c_{3} \end{matrix} | . | \begin{matrix}d_{1} & a_{1} & f_{1} \\ d_{2} & a_{2} & f_{2} \\ d_{3} & a_{3} & f_{3} \end{matrix} |

− | \begin{matrix}f_{1} & b_{1} & c_{1} \\ f_{2} & b_{2} & c_{2} \\ f_{3} & b_{3} & c_{3} \end{matrix} | . | \begin{matrix}d_{1} & e_{1} & a_{1} \\ d_{2} & e_{2} & a_{2} \\ d_{3} & e_{3} & a_{3} \end{matrix} | = | \begin{matrix}a_{1} & e_{1} & f_{1} \\ a_{2} & e_{2} & f_{2} \\ a_{3} & e_{3} & f_{3} \end{matrix} | . | \begin{matrix}d_{1} & b_{1} & c_{1} \\ d_{2} & b_{2} & c_{2} \\ d_{3} & b_{3} & c_{3} \end{matrix} | .

Denoting the column vectors of M by a,b,c,d,e,f, this identity written using tableaux takes the following form

\begin{matrix}a & b & c \\ d & e & f \end{matrix} = \begin{matrix}a & e & f \\ d & b & c \end{matrix} .

It is readily deduced from Turnbull’s identity, that we shall now state. Let τ be a tableau and denote by A the set of its boxed elements. Let B be the set of

elements of a given row, for example the first one, which do not belong to A. Finally let C denote the set of all other elements of τ. One can build another tableau υ equal to τ by merely exchanging the roles of A and B as shown by the following diagram:

τ =  B A C  =  A  B B C = ν

More precisely, we have

Theorem 2.1. (Turnbull’s identity, [Tu], p.209.) Let k be the number of elements of A.

(i) If k > n, τ = 0. (ii) If k ≤ n, and υ is built by

a. exchanging each element of A which does not belong to the first row, with any

element of B; b. boxing the elements of B; c. removing the boxes of the elements of A;

then τ = υ. For instance, taking n = 6, k = 4, A = {a,b,c,d}, B = {α,β,γ,δ,ε}, one has

τ = \begin{matrix}a & α & β & γ & δ & ε \\ b & c & f & g & h & i \\ d & j & k & l & m & o \end{matrix} = \begin{matrix}a & b & c & d & δ & ε \\ α & β & f & g & h & i \\ γ & j & k & l & m & o \end{matrix} = υ .

We shall now deduce from this identity some formulae for the power of a staircase Schur function.

3 Powers of a staircase Schur function and symmetric analogues of Bessel polynomials

Let ρ(k,m) denote the staircase partition (m, 2m, …, km). The n^{th} power of S_{ρ(k,m)} admits the following expression:

Theorem 3.1.

(S_{ρ(k,m)})^{n} = ∑_{I⊂ρ(n−1,m)}(−1)^{|ρ(n−1,m)|−|I|} S_{ρ(n+k−1,m)/I} Δ_{I} ,

where Δ_{I} is the minor taken on the lines i_{1} + 1, i_{2} + 2, …, i_{n−1} + n − 1 of the following (m + 1)(n − 1) × (n − 1) matrix:

M = ( \begin{matrix}S_{ρ(k−1,m)/1^{m}} & S_{ρ(k−1,m)/1^{2m+1}} & … & S_{ρ(k−1,m)/1^{(n−1)m+n−2}} \\ S_{ρ(k−1,m)/1^{m−1}} & S_{ρ(k−1,m)/1^{2m}} & … & S_{ρ(k−1,m)/1^{(n−1)m+n−3}} \\ ⋮ & ⋮ & & ⋮ \\ S_{ρ(k−1,m)} & S_{ρ(k−1,m)/1^{m+1}} & … & S_{ρ(k−1,m)/1^{(n−2)(m+1)}} \\ 0 & S_{ρ(k−1,m)/1^{m}} & … & S_{ρ(k−1,m)/1^{(n−2)m+n−3}} \\ 0 & S_{ρ(k−1,m)/1^{m−1}} & … & S_{ρ(k−1,m)/1^{(n−2)m+n−4}} \\ ⋮ & ⋮ & & ⋮ \\ 0 & S_{ρ(k−1,m)} & … & S_{ρ(k−1,m)/1^{(n−3)(m+1)}} \\ 0 & 0 & … & S_{ρ(k−1,m)/1^{(n−3)m+n−4}} \\ ⋮ & ⋮ & & ⋮ \\ 0 & 0 & … & S_{ρ(k−1,m)} \end{matrix} ) .

Note that we allow the first parts of I to be zero. Before proving the theorem, let us illustrate it by a few examples.

Example 3.2. We take n = 3, m = 1 and k = 4. The theorem states that

(S_{1234})^{3} = ∑_{I⊂12}(−1)^{3−|I|} S_{123456/I} Δ_{I} ,

where I runs over the set {00, 01, 02, 11, 12} and Δ_{I} is the minor taken on the lines i_{1} + 1, i_{2} + 2 of the 4 × 2 matrix:

M = ( \begin{matrix}S_{123/1} & S_{123/1^{3}} \\ S_{123} & S_{123/1^{2}} \\ 0 & S_{123/1} \\ 0 & S_{123} \end{matrix} ) = ( \begin{matrix}S_{1234/1112} & S_{1234/1114} \\ S_{1234/1111} & S_{1234/1113} \\ S_{1234/1110} & S_{1234/1112} \\ S_{1234/111(−1)} & S_{1234/1111} \end{matrix} ) .

The second expression of M shows more clearly the regularity of its columns (we recall that for any partition J, the determinants S_{J/1110} and S_{J/111(−1)} are null having two identical rows). Expanding this sum and the determinants Δ_{I} yields the following relation:

(S_{1234})^{3} = S_{123456} S_{123} S_{123/111} − S_{123456} S_{123/1} S_{123/11} + S_{123456/1} S_{123/1} S_{123/1} −S_{123456/11} S_{123} S_{123/1} − S_{123456/2} S_{123} S_{123/1} + S_{123456/12} S_{123} S_{123} .

Example 3.3. We take n = 2, so that theorem 3.1 reads

(S_{ρ(k,m)})^{2} = ∑_{0≤i≤m}(−1)^{m−i} S_{ρ(k+1,m)/i} S_{ρ(k−1,m)/1^{m−i}} ,

which is the formula investigated by Goulden [Go] for the square of a staircase Schur function.

Proof of Theorem 3.1. We shall apply Turnbull’s identity 2.1. Set N = k + (n − 1)(m + 1) and consider the N × ∞ matrix

M = ( \begin{matrix}S_{0} & S_{1} & S_{2} & S_{3} & … & 1 & 0 & 0 & … & 0 \\ 0 & S_{0} & S_{1} & S_{2} & … & 0 & 1 & 0 & … & 0 \\ 0 & 0 & S_{0} & S_{1} & … & 0 & 0 & 1 & … & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & & ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & 0 & 0 & … & 0 & 0 & 0 & … & 1 \end{matrix} ) .

Let 0, 1, 2, 3, … denote the column vectors of the left part of M, and 1^{∗}, 2^{∗}, 3^{∗}, … N^{∗}, denote the last N column vectors of M. For brevity we shall write m_{p} = pm + p − 1. The Schur function S_{ρ(k,m)} may be expressed as a maximal minor of M in n different ways

S_{ρ(k,m)} = [m_{1} m_{2} … m_{k} (k + 1)^{∗} (k + 2)^{∗} … N^{∗}]

= [0 1 … (m_{1} − 1) m_{1} m_{2} … m_{k+1} (k + m_{1} + 1)^{∗} (k + m_{1} + 2)^{∗} … N^{∗}]

⋮
= [0 1 2 … (m_{n−1} − 1) m_{n−1} m_{n} … m_{k+n−1}] ,

so that its n^{th} power is equal to the product of these n minors

(S_{ρ(k,m)})^{n} =

\begin{matrix}m_{1} & … & m_{k} & (k + 1)^{∗} & … & & & … & N^{∗} \\ 0 & … & (m_{1} − 1) & m_{1} & … & m_{k+1} & (k + m_{1} + 1)^{∗} & … & N^{∗} \\ ⋮ & & & & ⋮ & & & & ⋮ \\ 0 & … & & & … & (m_{n−1} − 1) & m_{n−1} & … & m_{k+n−1} \end{matrix} .

Now, recalling that a tableau having two equal letters on the same row is equal to zero, we may box the letters m_{1}, m_{2}, …,m_{k} in the first row, the letter m_{k+1} in the second row, …, the letter m_{k+n−1} in the n^{th} row, and apply Turnbull’s identity. This yields

(S_{ρ(k,m)})^{n}

= \begin{matrix}m_{1} & … & m_{k} & (k + 1)^{∗} & … & & & … & N^{∗} \\ 0 & … & (m_{1} − 1) & m_{1} & … & m_{k+1} & (k + m_{1} + 1)^{∗} & … & N^{∗} \\ ⋮ & & & & ⋮ & & & & ⋮ \\ 0 & … & & & … & (m_{n−1} − 1) & m_{n−1} & … & m_{k+n−1} \end{matrix}

= \begin{matrix}m_{1} & … & m_{k} & m_{k+1} & … & m_{k+n−1} & (k + n)^{∗} & … & N^{∗} \\ 0 & … & (m_{1} − 1) & m_{1} & … & (k + 1)^{∗} & (k + m_{1} + 1)^{∗} & … & N^{∗} \\ ⋮ & & & & ⋮ & & & & ⋮ \\ 0 & … & & & … & (m_{n−1} − 1) & m_{n−1} & … & (k + n − 1)^{∗} \end{matrix} .

This last tableau represents a sum of products of Schur functions of the following type

±S_{ρ(n+k−1,m)/I} S_{ρ(k−1,m)/1^{j_{1}}} … S_{ρ(k−1,m)/1^{j_{n−1}}} .

The coefficient of a given S_{ρ(n+k−1,m)/I} in this sum is in fact a tableau having n − 1 rows with only one letter boxed in each row, that is a determinant of order n − 1. There is no difficulty in recognizing this determinant as the minor Δ_{I} defined above.

The formula for the square of S_{ρ(k,1)} may be presented as the computation of a determinant of Schur functions

| \begin{matrix}S_{ρ(k−1,1)} & S_{ρ(k+1,1)} \\ S_{ρ(k−1,1)/1} & S_{ρ(k+1,1)/1} \end{matrix} | = (S_{ρ(k,1)})^{2} .  (8)

The same may be done for theorem 3.1, which has also a determinantal formulation. We shall write for brevity ρ(k) in place of ρ(k,1).

Theorem 3.4.

| \begin{matrix}S_{ρ(k−1)} & 0 & … & 0 & … & 0 & S_{ρ(k+1)} \\ S_{ρ(k−1)/1} & 0 & … & 0 & … & S_{ρ(k+2)} & S_{ρ(k+1)/1} \\ S_{ρ(k−1)/1^{2}} & S_{ρ(k,1)} & … & 0 & … & S_{ρ(k+2)/1} & S_{ρ(k+1)/1^{2}} \\ ⋮ & ⋮ & & ⋮ & & ⋮ & ⋮ \\ S_{ρ(k−1)/1^{2n−3}} & S_{ρ(k−1)/1^{2n−5}} & … & S_{ρ(k+n−1)/1^{n−1}} & … & S_{ρ(k+2)/1^{2n−4}} & S_{ρ(k+1)/1^{2n−3}} \end{matrix} |

= (S_{ρ(k)})^{n} S_{ρ(k+1)} S_{ρ(k+2)} … S_{ρ(k+n−2)} .

Example 3.5. For n = 3 and k = 4 one has

| \begin{matrix}S_{123} & 0 & 0 & S_{12345} \\ S_{123/1} & 0 & S_{123456} & S_{12345/1} \\ S_{123/1^{2}} & S_{123} & S_{123456/1} & S_{12345/1^{2}} \\ S_{123/1^{3}} & S_{123/1} & S_{123456/1^{2}} & S_{12345/1^{3}} \end{matrix} | = (S_{1234})^{3} S_{12345} .

Example 3.6. For n = 4 and k = 6 one has

| \begin{matrix}S_{12345} & 0 & 0 & 0 & 0 & S_{1234567} \\ S_{12345/1} & 0 & 0 & 0 & S_{12345678} & S_{1234567/1} \\ S_{12345/1^{2}} & S_{12345} & 0 & S_{123456789} & S_{12345678/1} & S_{1234567/1^{2}} \\ S_{12345/1^{3}} & S_{12345/1} & 0 & S_{123456789/1} & S_{12345678/1^{2}} & S_{1234567/1^{3}} \\ S_{12345/1^{4}} & S_{12345/1^{2}} & S_{12345} & S_{123456789/1^{2}} & S_{12345678/1^{3}} & S_{1234567/1^{4}} \\ S_{12345/1^{5}} & S_{12345/1^{3}} & S_{12345/1} & S_{123456789/1^{3}} & S_{12345678/1^{4}} & S_{1234567/1^{5}} \end{matrix} |

= (S_{123456})^{4} S_{1234567} S_{12345678} .

Proof. The proof of this last example will illustrate enough the principle of our computation. We consider the 12 × 30 matrix

M = ( \begin{matrix}S_{0} & S_{1} & S_{2} & S_{3} & … & S_{17} & 1 & 0 & 0 & … & 0 \\ 0 & S_{0} & S_{1} & S_{2} & … & S_{16} & 0 & 1 & 0 & … & 0 \\ 0 & 0 & S_{0} & S_{1} & … & S_{15} & 0 & 0 & 1 & … & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & & ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & 0 & 0 & … & S_{6} & 0 & 0 & 0 & … & 1 \end{matrix} ) ,

whose column vectors are denoted 0, 1, 2, … 17, 1^{∗}, 2^{∗}, … 12^{∗}. The determinant D of example 3.6 is represented by the following tableau

D = \begin{matrix}0 & 1 & 3 & 5 & 7 & 9 & 11 & 7^{∗} & 9^{∗} & 10^{∗} & 11^{∗} & 12^{∗} \\ 0 & 1 & 2 & 3 & 5 & 7 & 9 & 11 & 13 & 8^{∗} & 11^{∗} & 12^{∗} \\ 0 & 1 & 2 & 3 & 4 & 5 & 7 & 9 & 11 & 13 & 15 & 9^{∗} \\ 1 & 3 & 5 & 7 & 9 & 11 & 13 & 15 & 17 & 10^{∗} & 11^{∗} & 12^{∗} \\ 1 & 3 & 5 & 7 & 9 & 11 & 13 & 15 & 17 & 0 & 11^{∗} & 12^{∗} \\ 1 & 3 & 5 & 7 & 9 & 11 & 13 & 15 & 17 & 0 & 2 & 12^{∗} \end{matrix} .

Now we transform this tableau by Turnbull’s identity and note that all boxes may be removed because of the presence of identical letters

D = \begin{matrix}0 & 1 & 3 & 5 & 7 & 9 & 11 & 13 & 9^{∗} & 10^{∗} & 11^{∗} & 12^{∗} \\ 0 & 1 & 2 & 3 & 5 & 7 & 9 & 11 & 13 & 15 & 11^{∗} & 12^{∗} \\ 0 & 1 & 2 & 3 & 4 & 5 & 7 & 9 & 11 & 13 & 15 & 17 \\ 1 & 3 & 5 & 7 & 9 & 11 & 13 & 15 & 17 & 0 & 11^{∗} & 12^{∗} \\ 1 & 3 & 5 & 7 & 9 & 11 & 13 & 15 & 17 & 0 & 2 & 12^{∗} \\ 1 & 3 & 5 & 7 & 9 & 11 & 7^{∗} & 8^{∗} & 9^{∗} & 10^{∗} & 11^{∗} & 12^{∗} \end{matrix}

= \begin{matrix}0 & 1 & 3 & 5 & 7 & 9 & 11 & 13 & 9^{∗} & 10^{∗} & 11^{∗} & 12^{∗} \\ 0 & 1 & 2 & 3 & 5 & 7 & 9 & 11 & 13 & 15 & 11^{∗} & 12^{∗} \\ 0 & 1 & 2 & 3 & 4 & 5 & 7 & 9 & 11 & 13 & 15 & 17 \\ 1 & 3 & 5 & 7 & 9 & 11 & 13 & 15 & 17 & 0 & 11^{∗} & 12^{∗} \\ 1 & 3 & 5 & 7 & 9 & 11 & 13 & 15 & 17 & 0 & 2 & 12^{∗} \\ 1 & 3 & 5 & 7 & 9 & 11 & 7^{∗} & 8^{∗} & 9^{∗} & 10^{∗} & 11^{∗} & 12^{∗} \end{matrix}

= (S_{123456})^{4} S_{1234567}S_{12345678}

We shall end this section with an interpretation of theorem 3.4 in terms of orthogonal polynomials. First we shall give an immediate generalization of formula (8).

Proposition 3.7. For 1 ≤ p ≤ k + 1 we have

| \begin{matrix}S_{ρ(k−1)} & S_{ρ(k+1)} \\ S_{ρ(k−1)/1^{p}} & S_{ρ(k+1)/1^{p}} \end{matrix} | = S_{ρ(k)} S_{ρ(k)/1^{p−1}} ,

where ρ(k) is short for the partition 12 … k.

Proof. This is an instance of Plücker relations. We consider again the matrix M of the proof of theorem 3.1, taking N = k +2. The determinant to be computed is then represented by the 2 × (k + 2) tableau

(−1)^{p} \begin{matrix}0 & 1 & 3 & 5 & … & 2k − 1 & N^{∗} \\ 1 & 3 & 5 & 7 & … & 2k + 1 & (N − p)^{∗} \end{matrix} .

Applying Turnbull’s identity, this tableau is transformed into

(−1)^{p} \begin{matrix}0 & 1 & 3 & 5 & … & 2k − 1 & 2k + 1 \\ 1 & 3 & 5 & 7 & … & N^{∗} & (N − p)^{∗} \end{matrix}

= (−1)^{p} \begin{matrix}0 & 1 & 3 & 5 & … & 2k − 1 & 2k + 1 \\ 1 & 3 & 5 & 7 & … & N^{∗} & (N − p)^{∗} \end{matrix} = S_{ρ(k)} S_{ρ(k)/1^{p−1}} .

¿From this proposition we deduce the following result:

Theorem 3.8. Let 𝔼 be an alphabet and let us denote by π_{k} the polynomial of degree k in x with coefficients in the ring of symmetric functions of 𝔼: π_{k}(x) = S_{ρ(k)}(𝔼 − x). Then π_{k} satisfies the three term recurrence relationship (k ≥ 1)

S_{ρ(k−1)}(𝔼)π_{k+1}(x) + xS_{ρ(k)}(𝔼)π_{k}(x) − S_{ρ(k+1)}(𝔼)π_{k−1}(x) = 0 .

In other words (π_{k}(x))_{k≥0} is a family of orthogonal polynomials.

Proof. The expansion of π_{k} is obtained by considering it as a multi-Schur function and expanding this determinant along its last column. Indeed, by lemma 2.1,

π_{k}(x) = S_{12...k0}(𝔼, …,𝔼,x) = ∑_{0≤i≤k} S_{12...k/1^{i}}(𝔼)(−x)^{i} .

By linearity we deduce from proposition 3.7 that

| \begin{matrix}S_{ρ(k−1)} & S_{ρ(k+1)} \\ π_{k−1}(x) & π_{k+1}(x) \end{matrix} | = −xS_{ρ(k)} π_{k}(x) ,

which is the required relationship.

Thus, theorem 3.8 states that the elimination of x^{0} between π_{k−1}(x) and π_{k+1}(x) produces a polynomial proportional to xπ_{k}(x). As a consequence the elimination of x^{0}, x^{1}, x^{2} between π_{k−1}(x), π_{k+1}(x), xπ_{k+2}(x), x^{2}π_{k−1}(x) will produce a polynomial proportional to x^{3}π_{k}(x). This elimination is expressed by the following formula

| \begin{matrix}S_{ρ(k−1)} & 0 & 0 & S_{ρ(k+1)} \\ S_{ρ(k−1)/1} & 0 & S_{ρ(k+2)} & S_{ρ(k+1)/1} \\ S_{ρ(k−1)/1^{2}} & S_{ρ(k−1)} & S_{ρ(k+2)/1} & S_{ρ(k+1)/1^{2}} \\ π_{(k−1)}(x) & x^{2}π_{k−1}(x) & −xπ_{k+2}(x) & π_{k+1}(x) \end{matrix} | = −(S_{ρ(k)})^{2} S_{ρ(k+1)} x^{3}π_{k}(x).

Picking up the coefficient of x^{3} on each side we find again the case n = 3 of theorem 3.4. And this process may clearly be carried out for larger values of n.

The computation of the moments of the sequence of orthogonal polynomials (π_{k}(x))_{k≥0} will result from the following identity. We shall denote by ρ(n) + m the partition (1, 2, …, n − 1, n + m).

Theorem 3.9. Let k, n be integers such that 1 ≤ k ≤ n. There holds:

| \begin{matrix}S_{ρ(k)/1^{k−1}} & S_{ρ(k+1)/1^{k−1}} & … & S_{ρ(n)/1^{k−1}} \\ S_{ρ(k)/1^{k}} & S_{ρ(k+1)/1^{k}} & … & S_{ρ(n)/1^{k}} \\ ⋮ & ⋮ & & ⋮ \\ S_{ρ(k)/1^{n−1}} & S_{ρ(k+1)/1^{n−1}} & … & S_{ρ(n)/1^{n−1}} \end{matrix} | = S_{ρ(k−1)}P_{ρ(k)+(n−k+1)}S_{ρ(k+1)}S_{ρ(k+2)} … S_{ρ(n−1)} ,

where P_{ρ(k)+(n−k+1)} = ∑_{0≤2i≤n−k+1} Λ_{ρ(k)+2i}S_{n−k+1−2i}.

Example 3.10. For n = 6, k = 3 one has

| \begin{matrix}S_{123/1^{2}} & S_{1234/1^{2}} & S_{12345/1^{2}} & S_{123456/1^{2}} \\ S_{123/1^{3}} & S_{1234/1^{3}} & S_{12345/1^{3}} & S_{123456/1^{3}} \\ S_{123/1^{4}} & S_{1234/1^{4}} & S_{12345/1^{4}} & S_{123456/1^{4}} \\ S_{123/1^{5}} & S_{1234/1^{5}} & S_{12345/1^{5}} & S_{123456/1^{5}} \end{matrix} | = S_{12}P_{127}S_{1234}S_{12345} ,

where P_{127} = Λ_{123}S_{4} + Λ_{125}S_{2} + Λ_{127}.

Proof. The proof is similar to the proof of theorem 3.4, the only difference being that one has to apply Turnbull’s identity several times in order to obtain the required factorization.

Corrolary 3.11. The n^{th} moment of the sequence of orthogonal polynomials (π_{k}(x))_{k≥0} is the sum of all hook Schur functions of weight n + 1: μ(x^{n}) = ∑_{0≤i≤n} S_{1^{i}(n+1−i)}(𝔼) .

Proof. Let q_{k}(x) denote the monic polynomial q_{k}(x) = S_{ρ(k)}(𝔼 − x)/(−1)^{k}S_{ρ(k−1)}(𝔼). Theorem 3.9 may be seen as the resolution of the linear system in the unknown a_{nk}:

x^{n} = ∑_{k} a_{nk}q_{k}(x) .  (9)

The result is that a_{nk} = P_{ρ(k+1)+n−k}(𝔼)/S_{ρ(k+1)} . Denote by Sym(𝔼) the ring of symmetric functions of the alphabet 𝔼 and let μ be the functional on Sym(𝔼)[x] associated with the sequence of orthogonal polynomials (π_{k}(x))_{k≥0}. In fact μ is defined up to a constant factor, so that we can add the condition μ(1) = S_{1}(𝔼). Then, recalling that by definition μ(q_{k}(x)) = 0, k ≥ 1 and applying μ to the equality (9), one obtains that μ(x^{n}) = P_{n+1}(𝔼) = ∑_{0≤i≤n} S_{1^{i}(n+1−i)}(𝔼) , the last equality resulting easily from Pieri rule for the multiplication of a Schur function by an elementary symmetric function.

Remarks. (i) It can be shown that the symmetric function

∑_{0≤2i≤n−k+1} Λ_{ρ(k)+2i}S_{n−k+1−2i}

is nothing but the Schur P -function P_{ρ(k)+n−k+1}, hence the notation used above. More generally it has been shown in [LLT] that there exists for every Schur P -function a similar quadratic expansion in terms of ordinary Schur S-functions.

(ii) It is easily checked that S_{ρ(n)/1^{m}} = S_{ρ(n)/m}. It follows that the polynomials ϕ_{k}(x) = S_{ρ(k)}(𝔼 + x) are equal to the polynomials π_{k}(−x), and therefore also satisfy a three term recurrence relationship:

S_{ρ(k−1)}(𝔼)ϕ_{k+1}(x) − xS_{ρ(k)}(𝔼)ϕ_{k}(x) − S_{ρ(k+1)}(𝔼)ϕ_{k−1}(x) = 0 .

They are orthogonal for the moments ν(x^{n}) = (−1)^{n} ∑_{0≤i≤n} S_{1^{i}(n+1−i)}(𝔼) .

We must now emphasize on the differences between the two Schur function expressions of orthogonal polynomials hitherto encountered. As recalled in section 2, the family of orthogonal polynomials associated with a given functional μ may always be represented up to a constant factor by the sequence of rectangle Schur functions S_{n^{n}}(𝔼 − x), n ≥ 0, where 𝔼 is the alphabet formally defined by S_{k}(𝔼) = μ(x^{k}), k ≥ 0. In contrast Schur functions of the type S_{12...n}(𝔼 − x), n ≥ 0 or S_{12...n}(𝔼 + x), n ≥ 0 represent only particular families of orthogonal polynomials. Indeed, the recurrence relationship of theorem 3.8 is not generic, for the coefficient of π_{n}(x) in this relationship is of the type α_{n}x while the general form of this coefficient is α_{n}x + β_{n}. The following example shows that these polynomials may be seen as symmetric analogues of Bessel polynomials.

Example 3.12.

Let E be the alphabet formally defined by S_{k}(E) = 1/k!, k ≥ 0. It is a classical result that for any partition I of weight n, there holds S_{I}(E) = f_{I}/n!, where f_{I} is the number of standard Young tableaux of shape I, i.e. the dimension of the irreducible representation of 𝔖_{n} indexed by I. It follows that the moments associated to the sequence (ϕ_{k}(x))_{k≥0} are equal to ν(x^{n}) = (−2)^{n}/(n + 1)! , that is, are equal to the moments of Bessel polynomials (see for example [Ch]). The corresponding polynomials (suitably normalized)

y_{n}(x) = S_{12...n}(E + x)/S_{12...n}(E), n ≥ 0,  (10)

are therefore the Bessel polynomials. The first ones are

\begin{matrix}y_{0}(x) = & 1, \\ y_{1}(x) = & 1 + x, \\ y_{2}(x) = & 1 + 3x + 3x^{2}, \\ y_{3}(x) = & 1 + 6x + 15x^{2} + 15x^{3}, \\ y_{4}(x) = & 1 + 10x + 45x^{2} + 105x^{3} + 105x^{4}. \end{matrix}

Formula (10) provides the following determinantal expression

y_{n}(x) = ∏_{1≤i≤n}(2(n − i) + 1)^{i} | \begin{matrix}1/1! & 1/3! & 1/5! & … & 1/(2n − 1)! & (−x)^{n} \\ 1/0! & 1/2! & 1/4! & … & 1/(2n − 2)! & (−x)^{n−1} \\ 0 & 1/1! & 1/3! & … & 1/(2n − 3)! & (−x)^{n−2} \\ ⋮ & ⋮ & ⋮ & & ⋮ & ⋮ \\ 0 & & & & 1/n! & −x \\ 0 & & & & 1/(n − 1)! & 1 \end{matrix} | .

There exists a fairly extensive literature devoted to these polynomials. For a detailed account up to 1978, the interested reader is referred to [Gr]. We shall also mention the recent work of Dulucq and Favreau who have presented a combinatorial model for these polynomials, based upon weighted involutions [DF].

Note that our point of view provides at once a q-analogue for Bessel polynomials, by merely replacing E by the alphabet E_{q} = {1, q, q^{2}, …}. More precisely, setting v = x/(1 − q) we define

y_{n}(x,q) = q^{\binom{n}{2}}S_{12...n}(E_{q} + v)/S_{12...n}(E_{q}).

The y_{n}(x,q) are orthogonal polynomials in x, whose coefficients are polynomials in q. The first ones are

y_{0}(x,q) =1,
y_{1}(x,q) =1 + x,
y_{2}(x,q) =q + (1 + q + q^{2})x + (1 + q + q^{2})x^{2},
y_{3}(x,q) =q^{3} + q(1 + q^{2})(1 + q + q^{2})x + (1 + q + q^{2})(1 + q + q^{2} + q^{3} + q^{4})x^{2}
+ (1 + q + q^{2})(1 + q + q^{2} + q^{3} + q^{4})x^{3}.

These q-analogues are equal to those obtained by Dulucq by means of the combinatorial model of Dulucq and Favreau [D].

4 Expression of a symmetric polynomial in terms of the power sums of odd degree

Let 𝕏 denote a finite set of variables 𝕏 = {x_{1}, …,x_{n}}. It is well known that any symmetric polynomial F (𝕏) may be expressed as a polynomial function of the power sums ψ_{k}(𝕏) = ∑_{x∈𝕏} x^{k}, k = 1,2 … n. It was shown in the last century that F (𝕏) may also be expressed as a rational function of ψ_{k}(𝕏), k = 1,3,5, … 2n − 1 ([Bo], [La]). Pólya [Po] and Foulkes [Fo], among others, have proposed explicit expressions for some particular symmetric polynomials F (𝕏). For instance

Proposition 4.1. The k^{th} elementary symmetric function is equal to Λ_{k}(𝕏) = S_{ρ(n)/1^{n−k}}(𝕏)/S_{ρ(n−1)}(𝕏) , the right-hand side being a rational function of the power sums of odd degree.

Proof. For sake of completeness we sketch the proof. It is known that S_{ρ(n−1)}(𝕏) = ∏_{i<j}(x_{i} + x_{j}) . Taking an additionnal variable z we have also

S_{ρ(n)}(𝕏 + z) = ∏_{i<j}(x_{i} + x_{j}) ∏_{i}(x_{i} + z) = S_{ρ(n−1)}(𝕏) ∑_{k} z^{n−k}Λ_{k}(𝕏) .

Thus, comparing the coefficients of z^{n−k}, we get Λ_{k}(𝕏) = S_{ρ(n)/n−k}(𝕏)/S_{ρ(n−1)}(𝕏) = S_{ρ(n)/1^{n−k}}(𝕏)/S_{ρ(n−1)}(𝕏) . Finally, we recall that the staircase Schur function S_{ρ(n)} and all its derivatives S_{ρ(n)/I} depend only on the odd power sums ψ_{2p+1}, that is, belong to the subring generated by ψ_{1}, ψ_{3}, ψ_{5}, . . ..

Now theorem 3.9 provides at once a similar expression for the complete symmetric functions S_{k}(𝕏).

Proposition 4.2. The k^{th} complete symmetric function is equal to

S_{k}(𝕏) = \frac{∑_{0≤2i≤k} Λ_{ρ(n−1)+2i}(𝕏)S_{k−2i}(𝕏)}{S_{ρ(n−1)}(𝕏)} ,

the right-hand side being a rational function of the power sums of odd degree.

Proof. Theorem 3.9 shows that the function ∑_{0≤2i≤k} Λ_{ρ(n−1)+2i}S_{k−2i} may be expressed in terms of Schur functions of the type S_{ρ(m)/1^{p}}, which all belong to ℤ[ψ_{1},ψ_{3},ψ_{5}, …]. On the other hand, since 𝕏 is finite of cardinality n, Λ_{ρ(n−1)+2i}(𝕏) = 0 for all i > 0.

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