HALL-LITTLEWOOD FUNCTIONS AND KOSTKA-FOULKES POLYNOMIALS

IN REPRESENTATION THEORY

J. Désarménien, B. Leclerc and J.-Y. Thibon

Abstract

This paper presents a survey of recent applications of Hall-Littlewood functions and Kostka-Foulkes polynomials to the representation theory of the general linear group GL(n, C) and of the symmetric group S_{n}. The reviewed topics include the q-analogue of Kostant’s partition function, vertex operators, generalized exponents of GL(n, C) and S_{n}-harmonic polynomials. We also give a detailed description of the various combinatorial interpretations of Kostka-Foulkes polynomials. We conclude with the study of Hall-Littlewood functions at roots of unity, which provide a combinatorial description of certain plethysms.

Résumé

Cet article donne un aperçu de plusieurs applications récentes des fonctions de Hall-Littlewood et des polynômes de Kostka-Foulkes aux représentations du groupe linéaire GL(n, C) et du groupe symétrique S_{n}. Parmi les sujets abordés figurent notamment le q-analogue de la fonction de partition de Kostant, les opérateurs vertex, les exposants généralisés de GL(n, C) et les polynômes harmoniques de S_{n}. Nous donnons aussi une description détaillée des diverses interprétations combinatoires des polynômes de Kostka-Foulkes. Nous concluons par l’étude des fonctions de Hall-Littlewood aux racines de l’unité, qui fournit une interprétation combinatoire de certains pléthysmes.

1 Introduction

The original motivation for defining Hall polynomials, Hall algebra and Hall-Littlewood symmetric functions comes from group theory. It is well known that any finite Abelian p-group G is a direct sum of cyclic subgroups, of orders p^{λ_{1}}, p^{λ_{2}}, …, p^{λ_{r}}, say, where we may suppose that λ_{1} ≥ λ_{2} ≥ … ≥ λ_{r} > 0. The partition λ = (λ_{1}, …, λ_{r}) determines G to within isomorphism, and is called the type of G. The investigation of the lattice structure of finite Abelian p-groups leads to the problem of enumerating the chains of subgroups

1 = H_{0} ≤ H_{1} ≤ … ≤ H_{s} = G

in G of type λ, such that H_{i}/H_{i+1} has type μ^{i} for i = 1, 2, …, s. Let g_{μ^{1}...μ^{s}}^{λ}(p) denote the number of such chains. Hall showed [Ha] that (i) g_{μ^{1}...μ^{s}}^{λ}(p) is a polynomial function of p; (ii) the numbers g_{λμ}^{ν}(p) can be taken as the structure constants of an associative and

commutative algebra H(p), isomorphic to the algebra of symmetric functions;

(iii) the polynomial function g_{λ,μ}^{ν}(p) vanishes identically in p if and only if the corresponding multiplication constant for the Schur functions c_{λμ}^{ν} = ⟨s_{λ} s_{μ} , s_{ν}⟩ is zero;

(iv) the leading term of g_{λμ}^{ν}(p) is equal to c_{λμ}^{ν} p^{n(λ)−n(μ)−n(ν)}, where n(λ) = ∑_{i}(i − 1)λ_{i}.

In fact, it has been brought to light recently by Johnsen [Jo] that as early as 1900, Steinitz had arrived at the very same results in a forgotten note on the theory of Abelian groups [St].

The Hall algebra H(p) with generators u_{λ}(p) may be identified with the algebra of symmetric functions by sending the complete homogeneous symmetric function h_{n} onto the sum ∑_{|λ|=n} u_{λ}(p), thus introducing a new family of symmetric functions u_{λ}(p). Hall was able to show that the functions u_{1^{n}}(p) coincide up to a power of p with the elementary symmetric functions:

u_{1^{n}}(p) = p^{−n(n−1)/2}e_{n} ,

but it remained to Littlewood to obtain an explicit expression for the symmetric functions u_{λ}(p) for any partition λ. Defining P_{λ}(q) by formulas (15) and (16) below, Littlewood [Li1] found that

u_{λ}(p) = p^{−n(λ)}P_{λ}(p^{−1}) .  (1)

It turns out that the P_{λ}(q), called now Hall-Littlewood functions, interpolate between two fundamental bases of the algebra of symmetric functions, namely, the basis of Schur functions s_{λ} when q = 0, and the basis of monomial functions m_{λ} when q = 1. The entries K_{λμ} of the transition matrix from m_{μ} to s_{λ} are well known in representation theory: K_{λμ} is the multiplicity of the weight μ in an irreducible 𝔤𝔩 (n, C)-module with highest weight λ. These K_{λμ}, or Kostka numbers, have several combinatorial expressions, the most popular being that K_{λμ} is the number of Young tableaux of shape λ and weight μ. They can also be evaluated in terms of Kostant’s partition function P(α) (see Section 2 below). P(α) is defined as the number of ways of writing the vector α = (α_{1}, …, α_{n}) as a sum

α = ∑_{1≤i<j≤n} ν_{ij} ϵ_{ij} , m_{ij} ≥ 0

of positive roots ϵ_{ij} = (0, …, 1, …, −1, …, 0). A natural q-analogue P_{q}(α) is obtained by taking into account the ‘length’ ∑ ν_{ij} of each partition, which gives back by means of Kostant’s formula a q-analogue K_{λμ}(q) of the Kostka numbers K_{λμ}. These polynomials K_{λμ}(q) happen to be none other than the entries of the transition matrix from the basis P_{μ}(q) to the basis s_{λ}.

Apart from the Hall algebra, the best known applications of Hall-Littlewood functions include the character theory of finite linear groups [Gr], projective and modular representations of symmetric groups [S], [Mo2] and various topics in algebraic geometry [HS]

[He] [Lu1] [Lu2]. The aim of these notes is to present some applications to the ordinary representation theory of S_{n} and of GL(n, C), with emphasis on the combinatorial aspects.

In particular, the interpretation of Kostka-Foulkes polynomials in terms of the charge of Young tableaux [LS2] [Sc], and in terms of the quantum numbers of rigged configurations [KKR] [KR] are presented and illustrated by many examples. The notations for symmetric functions are those of [Mcd].

2 Raising operators and Kostant’s partition function

The Jacobi-Trudi determinant for Schur functions can be compactly expressed by means of raising operators

s_{λ} = ∏_{i<j}(1 − R_{ij}) h_{λ}  (2)

where R_{ij} h_{λ} := h_{R_{ij}(λ)} and R_{ij}(λ) := (…, λ_{i} + 1, …, λ_{j} − 1, …).

One would like to be able to invert formula (2), i.e. to write

h_{λ} = ∏_{i<j}(1 − R_{ij})^{−1}s_{λ} .  (3)

The simplest way to justify such a formal manipulation seems to adopt the following point of view about raising operators. Rather than acting on elements of Z^{n}, we prefer to use linear maps from the group algebra Z[Z^{n}] to some commutative ring R.

It is customary to use as a basis of Z[Z^{n}] the formal exponentials (e^{α})_{α∈Z^{n}}, satisfying the relations e^{α}e^{β} = e^{α+β}. We shall follow this practice, and furthermore introduce n independent indeterminates x_{1}, …, x_{n} in order to identify Z[Z^{n}] with the ring of polynomials Z[x_{1}, x_{1}^{−1}, …, x_{n}, x_{n}^{−1}], by setting e^{α} = x_{1}^{α_{1}} ⋯ x_{n}^{α_{n}}. We set X = {x_{1}, … x_{n}} and X^{∨} = {x_{1}^{−1}, … x_{n}^{−1}}.

Now, let f be any Z-module homomorphism from Z[Z^{n}] to a commutative ring R. We define g = R_{ij}f by

g(e^{λ}) := f (\frac{x_{i}}{x_{j}} e^{λ}) .  (4)

In the applications we have in mind, R will be the ring Sym of symmetric functions, or of symmetric polynomials in n variables. Introducing in both cases the two linear maps

s : e^{λ} ⟼ s(λ) = s_{λ} and h : e^{λ} ⟼ h(λ) = h_{λ}

(with h_{i} = 0 for i < 0) and the element

θ_{n} = ∏_{1≤i<j≤n} (1 − \frac{x_{i}}{x_{j}} )

we see that (2) can be written

s(e^{λ}) = h(θ_{n}e^{λ}) ,

from which it follows immediately by linearity that

h(e^{λ}) = s(θ_{n}^{−1}e^{λ})

which is exactly formula (3). We thus see that this formula has to be interpreted as an identity between linear maps, h = ∏_{i<j}(1 − R_{ij})^{−1}s, evaluated at e^{λ}.

Consider now the function P : Z^{n} ⟶ N defined by

θ_{n}^{−1} = ∏_{i<j} (1 − \frac{x_{i}}{x_{j}} )^{−1} = ∑_{α∈Z^{n}} P(α)e^{α} .  (5)

This is an example of a vector partition function. It counts the number of ways a vector α can be expressed as a sum of the vectors

ϵ_{ij} := ϵ_{i} − ϵ_{j} (i < j)

where ϵ_{i} = (0, …, 1, …, 0) is the canonical basis of Z^{n}.

The vectors ϵ_{ij} (i ≠ j) are nothing but the root system A_{n−1}, and those ϵ_{ij} with i < j are generally taken as the positive roots. The sublattice P = {α| ∑_{i} α_{i} = 0} of Z^{n} can be identified with the weight lattice of 𝔰𝔩 (n, C), and for α ∈ P, P(α) is the number of ways in which the weight α can be expressed as a sum of positive roots. Thus, P is called Kostant’s partition function for the root system A_{n−1}. The partition function can of course be defined for any root system, and much of the theory presented in this paper can be extended to this general setting.

Let us return to (3). Using (5), we can rewrite it as

h_{μ} = ∑_{α∈Z^{n}} P(α)s_{μ+α}  (6)

where the Schur functions s_{μ+α} have to be standardized, i.e. expressed in terms of Schur functions indexed by partitions. Defining ρ = (n − 1, n − 2, …, 1, 0), we see that s_{μ+α} is equal to ε(w)s_{λ} if there exist a permutation w ∈ S_{n} and a partition λ such that μ + α + ρ = w(λ + ρ), and is 0 otherwise. Thus,

h_{μ} = ∑_{λ} ( ∑_{w∈S_{n}} ε(w)P (w(λ + ρ) − (μ + ρ))) s_{λ} .  (7)

On the other hand, the expansion of h_{μ} on the basis s_{λ} is well known to be given by the Kostka numbers

K_{λμ} = ⟨h_{μ} , s_{λ}⟩ = |Tab (λ, μ)|

where Tab (λ, μ) is the set of Young tableaux with shape λ and weight μ. Thus,

K_{λμ} = ∑_{w∈S_{n}} ε(w)P (w(λ + ρ) − (μ + ρ)) .  (8)

This formula is Kostant’s weight multiplicity formula (for GL(n, C)). Since one has as well

s_{λ} = ∑_{μ} K_{λμ}m_{μ}  (9)

it expresses the multiplicity K_{λμ} of the weight μ in the irreducible polynomial representation V_{λ} of GL(n, C) with highest weight λ.

Example 2.1 We put n = 4, λ = (3, 3, 2, 0), μ = (2, 2, 2, 2). Then λ + ρ = (6, 5, 3, 0) and μ + ρ = (5, 4, 3, 2). There are only three nonzero summands in (8), as shown in the following table.

\begin{matrix}w(λ + ρ) & w(λ + ρ) − (μ + ρ) & P(w(λ + ρ) − (μ + ρ)) & ε(w) \\ (6, 5, 3, 0) & (1, 1, 0, −2) & 7 & + \\ (5, 6, 3, 0) & (0, 2, 0, −2) & 3 & − \\ (6, 3, 5, 0) & (1, −1, 2, −2) & 1 & − \end{matrix}

Hence, K_{λμ} = 7 − 3 − 1 = 3. The corresponding three tableaux are

1 1 2 2 3 4 3 4  ^{1 1 2 2 3 3 4 4}  ^{1 1 3 2 2 4 3 4}

3 Hall-Littlewood functions and the q-analogue of the partition function

A natural statistics on ordinary partitions λ of a number n is the length (number of parts). That is, the generating function

∑_{n≥0} p(n)t^{n} = ∏_{k≥1}(1 − t^{k})^{−1}

can be refined in

∑_{m,n} p(n; m)q^{m}t^{n} = ∏_{k≥1}(1 − qt^{k})^{−1}

where p(n; m) is the number of partitions of n in m parts. The same can be done with vector partition functions, and the length q-analog of Kostant’s partition function has indeed been considered by several authors [Ka][Mcd][Lu1][Gu1]. For the root system A_{n−1}, it is given by

θ_{n}(q)^{−1} := ∏_{1≤i<j≤n} (1 − q \frac{x_{i}}{x_{j}} )^{−1} = ∑_{α∈Z^{n}} P_{q}(α)e^{α}  (10)

the coefficient of q^{k} in P_{q}(α) being the number of ways the weight α can be expressed as a sum of k positive roots.

With this at hand, it is then tempting to define a q-analog of the symmetric function h_{λ} by

Q′_{λ}(X; q) := ∏_{i<j}(1 − qR_{ij})^{−1}s_{λ}(X)  (11)

(the choice of this notation is explained below). One has then

Q′_{μ} = ∑_{λ} K_{λμ}(q)s_{λ}  (12)

where

K_{λμ}(q) = ∑_{w∈S_{n}} ε(w)P_{q} (w(λ + ρ) − (μ + ρ)) .  (13)

Example 3.1 Take λ = (3, 3, 2, 0), μ = (2, 2, 2, 2), as in Example 2.1. The computation of K_{λμ}(q) takes the form

\begin{matrix}w(λ + ρ) & w(λ + ρ) − (μ + ρ) & P_{q}(w(λ + ρ) − (μ + ρ)) & ε(w) \\ (6, 5, 3, 0) & (1, 1, 0, −2) & q^{2} + 3q^{3} + 2q^{4} + q^{5} & + \\ (5, 6, 3, 0) & (0, 2, 0, −2) & q^{2} + q^{3} + q^{4} & − \\ (6, 3, 5, 0) & (1, −1, 2, −2) & q^{3} & − \end{matrix}

Hence, K_{λμ}(q) = q^{3} + q^{4} + q^{5}.

It turns out, although this is by no means obvious, that the polynomials K_{λμ}(q) have nonnegative coefficients [LS2]. They are the Kostka-Foulkes polynomials, and the symmetric functions Q′_{λ} are, up to a simple transformation, the Hall-Littlewood functions.

To be more precise, recall that in the λ-ring formalism, the symmetric functions of the argument (1 − q)X are defined by

p_{k}((1 − q)X) = (1 − q^{k})p_{k}(X)

and similarly

p_{k} (\frac{X}{1 − q} ) = (1 − q^{k})^{−1}p_{k}(X) .

Then, the usual Hall-Littlewood functions are

Q_{μ}(X; q) = Q′_{μ}((1 − q)X; q) .  (14)

Littlewood’s original definition [Li1] of the Q-functions was, for X = {x_{1}, …, x_{n}}

Q_{λ}(X; q) = \frac{(1 − q)^{ℓ(λ)}}{[m_{0}]_{q}!} ∑_{w∈S_{n}} w (e^{λ} \frac{θ_{n}(q)}{θ_{n}(1)})  (15)

where m_{0} = n − ℓ(λ) and

[m_{0}]_{q}! = [m_{0}]_{q} [m_{0} − 1]_{q} ⋯ [1]_{q} = ∏_{1≤i≤m_{0}} \frac{1 − q^{i}}{1 − q} .

The normalization factor [m_{0}]_{q}!^{−1} is introduced in order to have stability with respect to adjunction of variables, i.e.

Q_{λ}(x_{1}, …, x_{n}; q) = Q_{λ}(x_{1}, …, x_{m+n}; q)|_{x_{m+1}=...=x_{m+n}=0} .

The coefficients of Q_{λ}(X; q) on the basis of Schur functions are polynomials in q having common factors of the type 1 − q^{i}. Hence one defines also the symmetric function

P_{λ}(X; q) = \frac{1}{(1 − q)^{ℓ(λ)} [m_{1}]_{q}! ⋯ [m_{n}]_{q}!} Q_{λ}(X; q) ,  (16)

which still has polynomial coefficients on the basis (s_{λ}). Littlewood showed that the symmetric functions obtained by Schur as generating functions for the projective characters of S_{n} were the Q_{λ}(X; −1). He also derived the formula

Q_{λ}(X; q) = ∏_{i<j}(1 − qR_{ij})^{−1}s_{λ}((1 − q)X)  (17)

which is easily seen, in view of definition (11), to be equivalent to (14). Finally, he pointed out that Hall’s functions u_{λ} were given by (1). This relation is established by Morris in [Mo3]. From our point of view, (P_{λ}(X; q)) is simply the adjoint basis of (Q′_{λ}(X; q)) for the usual scalar product of Sym.

4 Morris’ recurrence formula, vertex operators and Green polynomials

An important consequence of (17) is the following formula of Morris [Mo1], which allows to express a Kostka-Foulkes polynomial K_{λμ}(q) in terms of the lower weight ones K_{αν}(q), where ν = (μ_{2}, μ_{3}, …, μ_{s}) is obtained from μ by removing its largest part:

Q_{μ}(X; q) = ∑_{r≥0} q^{r} ∑_{α} K_{αν}(q) ∑_{β}⟨s_{β}h_{r} , s_{α}⟩s_{(μ_{1}+r,β)}((1 − q)X) .  (18)

This formula looks simpler when stated in terms of vertex operators. Recall that for any f ∈ Sym, there is a differential operator D_{f} on Sym (the Foulkes derivative) defined as the adjoint of the multiplication operator g ↦ fg, that is

⟨fg , h⟩ = ⟨g , D_{f}h⟩ .

Introducing the generating series

σ_{z}(X) := ∑_{r≥0} z^{r}h_{r}(X) and λ_{z}(X) := ∑_{r≥0} z^{r}e_{r}(X) ,  (19)

the generating series for the Schur functions indexed by vectors of the form (r, λ) (where λ is a fixed partition) can then be expressed as

∑_{r∈Z} z^{r}s_{(r,λ)} = σ_{z}D_{λ_{−1/z}}s_{λ} .  (20)

This identity is established by expanding the Jacobi-Trudi determinant by its first row, which causes the appearance of the skew Schur functions s_{λ/(1^{k})} = D_{e_{k}}s_{λ}. The operator

Γ_{z} = σ_{z} D_{λ_{−1/z}}  (21)

is a typical example of the so-called vertex operators of Mathematical Physics. It has been applied to the determination of the rational solutions of various hierarchies of soliton equations and to the construction of representations of certain infinite dimensional Lie algebras (see e.g. Kac’s book [Kac] Chap. 14 for examples and references). Some applications to symmetric functions and group characters can be found in [T][CT][ST]. Remark that in λ-ring notation,

D_{σ_{z}}f (X) = f(X + z) and D_{λ_{−z}}f (X) = f(X − z)

when z is an element of rank one (which means that e_{r}(z) = 0 for r > 1), so that

Γ_{z}f (X) = σ_{z}(X) f (X − \frac{1}{z}) .

Morris’ formula can now be rewritten as a generating function identity similar to (20) for the Q′-functions:

∑_{p∈Z} Q′_{(p,ν)}(X; q) = Γ_{1} Q′_{ν}(X + q; q) .  (22)

Indeed, replacing Q by Q′ in (18) and summing over p = μ_{1}, one gets

∑_{p∈Z} Q′_{(p,ν)} = ∑_{r≥0} q^{r} ∑_{α} K_{αν}(q) ∑_{β}⟨s_{β}h_{r} , s_{α}⟩ ∑_{m∈Z} s_{(m,β)}

= Γ_{1} {∑_{α} K_{αν}(q) ∑_{β} ⟨∑_{r≥0} q^{r}s_{α/r} , s_{β}⟩ s_{β}}

= Γ_{1} {∑_{α} K_{αν}(q)s_{α}(X + q)} = Γ_{1} Q′_{ν}(X + q) = Γ_{1}D_{σ_{q}}Q′_{ν} .

By homogeneity, one has as well

∑_{p∈Z} z^{p}Q′_{(p,ν)} = Γ_{z}Q′_{ν}(X + q) .  (23)

Example 4.1 Let us compute the stable Schur expansion of the functions Q′_{(p,2,1,1)}. First, we have

∑_{p∈Z} Q′_{p11} = Γ_{1} Q′_{11}(X + q) = Γ_{1} [q s_{2}(X + q) + s_{11}(X + q)]

= Γ_{1} [q (s_{2} + q s_{1} + q^{2}) + s_{11} + q s_{1}] = Γ_{1} [q s_{2} + (q + q^{2}) s_{1} + s_{11} + q^{3}s_{0}]

so that Q′_{211} = q s_{22} + (q + q^{2}) s_{31} + s_{211} + q^{3}s_{4}. Now,

∑_{p∈Z} Q′_{p211} = Γ_{1} [q s_{22}(X + q) + (q + q^{2}) s_{31}(X + q) + s_{211}(X + q) + q^{3}s_{4}(X + q)]

= Γ_{1} [q^{7}s_{0} + (q^{4} + q^{5} + q^{6}) s_{1} + (2q^{3} + q^{4} + q^{5}) s_{2} + (q^{2} + q^{3} + q^{4}) s_{11} + (q^{2} + q^{3} + q^{4}) s_{3}

+q s_{111} + (q + 2q^{2} + q^{3}) s_{21} + q^{3} s_{4} + (q + q^{2}) s_{31} + q s_{22} + s_{211}] .

Equation (22) provides a q-analogue of the character polynomials of the symmetric groups (see e.g. [Ke]) for the Green polynomials. The (renormalized) Green polynomials X_{λ}^{μ}(q) are q-analogues (at q = 0) of the characters χ_{λ}^{μ} of the symmetric group, defined by

X_{λ}^{μ}(q) = ⟨Q′_{μ} , p_{λ}⟩ .  (24)

If μ = (p, ν), it follows from (22) that

X_{λ}^{(p,ν)}(q) = ⟨σ_{1}D_{λ_{−1}}Q′_{ν}(X + q) , p_{λ}⟩
= ⟨Q′_{ν}(X + q − 1) , p_{λ}(X + 1)⟩
= ⟨Q′_{ν}(X + q − 1) , ∏_{i}(1 + p_{i})^{m_{i}(λ)}⟩  (25)

which yields an expression of X_{λ}^{μ}(q) as a polynomial Ξ^{ν}(q; m_{1}, m_{2}, …, m_{n}) in the multiplicities m_{i} of the parts of λ, which is independent of the first part μ_{1} of μ. The polynomials Ξ^{ν}(0 ; m_{1}, m_{2}, …, m_{n}) are the usual character polynomials of the symmetric group.

Example 4.2 Let λ = (1^{m_{1}}2^{m_{2}} ⋯ n^{m_{n}}). Then,

X_{λ}^{(p,1)}(q) = ⟨s_{1}(X + q − 1) , ∏_{i}(1 + p_{i})^{m_{i}}⟩

= ⟨p_{1} + (q − 1) , 1 + m_{1}p_{1} + ⋯⟩ = (q − 1) + m_{1}

and

X_{λ}^{(p,2)} = ⟨Q′_{2}(X + q − 1) , ∏_{i}(1 + p_{i})^{m_{i}}⟩

= ⟨s_{2}(X + q − 1) , (1 + m_{1}p_{1} + \binom{m_{1}}{2}p_{1}^{2} + ⋯)(1 + m_{2}p_{2} + ⋯) ⋯⟩

= q(q − 1) + (q − 1)\binom{m_{1}}{1} + \binom{m_{1}}{2} + \binom{m_{2}}{1} .

5 Generalized exponents for GL(n, C)

Consider the adjoint representation of G := GL(n, C), i.e. its action by conjugation on its Lie algebra 𝔤 = 𝔤𝔩 (n, C) = M_{n}(C)

Ad (g)A = gAg^{−1} .

Since as a G-module 𝔤 = V ⊗V^{∗}, where V = C^{n} is the defining representation of GL(n, C), with character ch_{V} = ∑_{i} x_{i} (the formal character ch_{U} of a polynomial representation can be viewed as the trace of the image of the diagonal matrix diag (x_{1}, …, x_{n}) in this representation), the character of the adjoint representation is

ch 𝔤 = (∑_{i=1}^{n} x_{i}) (∑_{j=1}^{n} x_{j}^{−1}) = ∑_{i,j} \frac{x_{i}}{x_{j}} .

Consider now the symmetric algebra

S^{•}(𝔤) = ⨁_{k≥0} S^{k}(𝔤)

as a graded G-module. The character of S^{k}(𝔤) is then given by the generating series

∑_{k≥0} q^{k}ch_{S^{k}(𝔤)} = σ_{q}(XX^{∨}) = ∏_{i,j} (1 − q \frac{x_{i}}{x_{j}} )^{−1} .  (26)

Let I = S^{•}(𝔤)^{G} be the algebra of G-invariants in S^{•}(𝔤). By a theorem of Kostant [Ko], S^{•}(𝔤) is a free module over I, generated by the G-harmonic polynomials, which are those elements of S^{•}(𝔤) annihilated by all G-invariant differential operators without constant term. One has S^{•}(𝔤) = I ⊗ H, where H = ⨁_{k≥0} H^{k} is the graded module of harmonic polynomials. Kostant defined the generalized exponents of a finite-dimensional G-module E to be the exponents e_{1}, …, e_{s} of the nonzero terms in the polynomial

F_{q}(E) := ∑_{k≥0}⟨E , H^{k}⟩q^{k} = ∑_{i=1}^{s} c_{i}q^{e_{i}}  (27)

where ⟨E , H^{k}⟩ = dim_{C}Hom_{G}(E , H^{k}) is the intertwining number.

The problem is to compute F_{q}(V_{λ}) for the irreducible representations V_{λ} of G. Here, we have to consider rational irreducible representations of GL(n, C), whose characters are of the form

e_{n}(X)^{m}s_{μ}(X)

where e_{n}(X) = x_{1} ⋯ x_{n}, m ∈ Z, and μ is a partition which can be taken of length ≤ n−1 since with n variables e_{n}^{m}(X)s_{μ}(X) = s_{μ+(m^{n})}(X). For GL(n, C), the result is

Theorem 5.1 The generating series of the characters of harmonic polynomials is equal to

∑_{k≥0} q^{k}ch_{H^{k}} = (q)_{n} ∏_{i,j} (1 − q \frac{x_{i}}{x_{j}} )^{−1} = ∑_{k≥0} e_{n}(X)^{−k}Q′_{(k^{n})}(X)  (28)

so that for a decreasing vector α with |α| = 0, such that λ = α + (k^{n}) is a partition, one has

F_{q}(V_{α}) = K_{λ,(k^{n})}(q) .  (29)

Several sophisticated proofs of this result appear in the literature. An elementary one, due to Gupta [Gu1], can be presented as follows.

Let Ω_{n} be the operator which sends the monomial e^{λ} onto the Schur function s_{λ}(X), i.e.

Ω_{n}(f) = ∑_{w∈S_{n}} w (f ∏_{i<j} (1 − \frac{x_{j}}{x_{i}} )^{−1}) .  (30)

It is clear that when f is symmetric, Ω_{n}(fg) = fΩ_{n}(g). Now, equation (26) can be written

(1 − q)^{n} ∏_{i<j} (1 − q \frac{x_{j}}{x_{i}} ) · σ_{q}(XX^{∨}) = ∏_{i<j} (1 − q \frac{x_{i}}{x_{j}} )^{−1} ,

and we can apply Ω_{n} to both sides. An easy induction shows that

Ω_{n} ∏_{i<j} (1 − q \frac{x_{j}}{x_{i}} )^{−1} = [n]_{q}! ,

and the r.h.s. becomes

Ω_{n} ( ∑_{α∈Z^{n}} P_{q}(α)e^{α}) = ∑_{α∈Z^{n}} P_{q}(α)s_{α}(X) = ∑_{γ↓} K_{γ,(0^{n})}(q)s_{γ}(X) ,

(sum over decreasing vectors of integers γ) so that

∑_{k≥0} q^{k}ch_{H^{k}} = (q)_{n}σ_{q}(XX^{∨}) = ∑_{k≥0}(x_{1} ⋯ x_{n})^{−k}Q′_{(k^{n})}(X) ,

as required.

6 Harmonic polynomials for the symmetric group

The modified Hall-Littlewood functions Q′_{λ} also appear as graded characteristics of the representations of the symmetric group in certain spaces of polynomials. Recall that the Frobenius characteristic is the linear map F from the representation ring R(S_{n}) to the ring of symmetric functions which sends the irreducible representation indexed by a partition λ to the Schur function s_{λ}. If χ is the character of a representation V, one has (cf. [Mcd])

F(V) = ∑_{|α|=n} χ(α)\frac{p_{α}}{z_{α}} .

For a graded module V = ⨁_{k} V_{k}, one defines the graded characteristic by

F_{q}(V) = ∑_{k} q^{k}F(V_{k}) .

For example, if V = C[x_{1}, …, x_{n}] is the ring of polynomials endowed with its usual graduation, the representation of S_{n} in V_{1} is the representation by permutation matrices w ↦ g_{w}, and the representation in V_{k} is its k-th symmetric power w ↦ S^{k}(g_{w}). Let w be a permutation of type α = (1^{m_{1}}2^{m_{2}} ⋯ n^{m_{n}}). Then, by a classical identity

∑_{k≥0} q^{k} tr S^{k}(g_{w}) = \frac{1}{det(I − q g_{w})} = ∏_{i≥1} \frac{1}{(1 − q^{i})^{m_{i}}} = p_{α} (\frac{1}{1 − q} )

(in λ-ring notation), so that

F_{q}(V) = ∑_{|α|=n} p_{α} (\frac{1}{1 − q} ) \frac{p_{α}(X)}{z_{α}} = h_{n} (\frac{X}{1 − q} ) .  (31)

The harmonic polynomials for the symmetric group S_{n} are those polynomials which are annihilated by all differential operators of the form

D = f (\frac{∂}{∂x_{1}} , …, \frac{∂}{∂x_{n}} )

where f is a symmetric polynomial without constant term. For example the Vandermonde determinant

Δ(X) = ∏_{r<s}(x_{r} − x_{s})

is harmonic, since for w ∈ S_{n},

wDΔ = DwΔ = ε(w)Dw

so that DΔ is an alternating polynomial of degree < degΔ. This implies that DΔ = 0 since any alternating polynomial is divisible by Δ. As a consequence, any partial derivative

∂^{α}Δ = \frac{∂^{α_{1}}}{∂x_{1}} … \frac{∂^{α_{n}}}{∂x_{n}} Δ

of Δ is harmonic since

D∂^{α}Δ = ∂^{α}DΔ = 0 .

A result of Chevalley (valid for any finite reflection group) shows that as an S_{n}-module,

C[x_{1}, …, x_{n}] = I ⊗ H  (32)

where I = C[x_{1}, …, x_{n}]^{S_{n}} is the ring of invariants, and H the module of harmonic polynomials. Since one clearly has

F_{q}(I) = ∑_{k≥0} q^{k}dim I_{k} h_{n}(X) = \frac{1}{(q; q)_{n}} h_{n}(X) ,

it follows that

F_{q}(H) = (q; q)_{n}h_{n} (\frac{X}{1 − q} ) = ω (Q′_{(1^{n})}(X; q))  (33)

where ω is the involution h_{n} ↔ e_{n}.

It can be shown that the other Q′ functions also arise as graded characteristics of natural submodules of H (up to a simple transformation) . However, this is far more complicated. Let us set

Q̃′_{μ}(X; q) = ∑_{λ} K̃_{λμ}(q)s_{λ}(X) = q^{n(μ)}Q′_{μ}(X; q^{−1})

and

F_{μ}(X; q) = ω(Q̃′_{μ}(X; q)) .

(In the special case when μ = (1^{n}) we have ω(Q′_{(1^{n})}) = Q̃′_{(1^{n})}, so that F_{(1^{n})} = Q′_{(1^{n})}). These symmetric functions first appeared as graded S_{n}-characteristics in algebraic geometry ([HS], see also [Mcd] ex. 9 p. 136). To be precise, if we denote by X_{μ} the submanifold of the flag manifold F (C^{n}) formed by the flags which are fixed by a unipotent u such that the nilpotent u − I has its Jordan canonical form described by the partition μ, there is an action of S_{n} on the cohomology ring H^{∗}(X_{μ}, Q) such that

F\sqrt{q}(H^{∗}(X_{μ})) = ∑_{i≥0} q^{i}F(H^{2i}(X_{μ})) = F_{μ}(X; q) .

It is well known that the cohomology ring of the full flag manifold F (C^{n}) = X_{(1^{n})} is isomorphic as an S_{n}-module to the ring of coinvariants (for the action σ f = f(x_{σ(1)}, …, x_{σ(n)}) which is conjugate to the above one)

Q[x_{1}, …, x_{n}]/I^{+}  (34)

where I^{+} is the ideal generated by the symmetric polynomials without constant term, and that this ring is itself isomorphic to the module H_{Q} of harmonic polynomials (over Q here).

A description of H^{∗}(X_{μ}) as the quotient of the ring of polynomials by a certain ideal is given in [DCP], and an isomorphic module of harmonic polynomials is constructed in [La]. Let λ be a partition of n and μ = λ′ = (μ_{1}, μ_{2}, …, μ_{k}) be its conjugate. The Specht polynomial Δ_{λ} is by definition the product of Vandermonde determinants

Δ_{λ} = ∏_{i=1}^{k} Δ(X^{(i)})

where X^{(i)} = {x_{r}|μ_{0} + ⋯ + μ_{r−1} < r ≤ μ_{0} + ⋯ + μ_{r}} (μ_{0} := 0) and for A ⊆ X

Δ(A) = ∏_{x_{r},x_{s}∈A, r<s}(x_{r} − x_{s}) .

Specht’s result states that the linear span of the orbit of Δ_{λ} under S_{n} is a model of the irreducible representation of S_{n} indexed by the partition λ. Now, one can also consider the linear span L_{λ} of the orbit of Δ_{λ} under the affine symmetric group, which is generated by the permutations and the translations T_{i} : x_{i} ↦ x_{i} + 1. The elements of this orbit are harmonic, since the Specht polynomials are easily seen to be (up to a scalar factor) partial derivatives of the full Vandermonde Δ(X). By Taylor’s formula, the image of a Specht polynomial under an element of the affine symmetric group is itself a linear combination of partial derivatives of the Vandermonde Δ(X) (which in fact span H). One can then state [La]:

Theorem 6.1 The linear span L_{λ} of the orbit of a Specht polynomial Δ_{λ} under the affine symmetric group is a submodule of H having as graded characteristic the modified Hall-Littlewood function Q̃′_{λ}(X; q).

Example 6.2 We choose λ = (2, 1, 1). The Specht polynomial Δ_{211} may be expressed as a derivative of the full Vandermonde Δ = Δ(x_{1}, …, x_{4}):

Δ_{211} = −\frac{1}{6} ∂^{(0,0,0,3)}Δ .

The homogeneous component of degree 3 of L_{211} is the Specht module W_{211}. A basis is for example

{∂^{(0,0,0,3)}Δ, ∂^{(0,0,3,0)}Δ, ∂^{(0,3,0,0)}Δ} .

The degree 2 component has the basis

{∂^{(0,1,0,3)}Δ, ∂^{(0,1,3,0)}Δ, ∂^{(0,3,0,1)}Δ, ∂^{(0,0,1,3)}Δ, ∂^{(0,0,3,1)}Δ} ,

and is isomorphic to the direct sum W_{22} ⊕ W_{31}. The degree 1 component has the basis

{∂^{(0,1,1,3)}Δ, ∂^{(1,0,1,3)}Δ, ∂^{(1,1,3,0)}Δ} ,

and is isomorphic to W_{31}. Finally, the degree 0 component is generated by ∂^{(0,1,2,3)}Δ. The graded characteristic is therefore

q^{3} s_{211} + (q + q^{2}) s_{31} + q^{2} s_{22} + s_{4} = q^{3} Q′_{211}(q^{−1}) = Q̃′_{211} .

Note that the interpretation of the F_{μ}(X; q) as graded characteristics yields the positivity of the Kostka-Foulkes polynomials.

7 Kostka-Foulkes polynomials and the charge of Young tableaux

In [Fo1], Foulkes asked for a combinatorial explanation of the positivity of Kostka polynomials. He conjectured the existence of a natural statistics c(t) on the set of Young tableaux such that

K_{λμ}(q) = ∑_{t∈Tab (λ,μ)} q^{c(t)} .

This conjecture was settled by Lascoux and Schützenberger [LS2], [Sc], who identified c(t) to be the rank of a poset structure on Tab (·, μ).

The definition of the charge c(t) rests upon the interpretation of Young tableaux as elements of a monoid, whose definition and basic properties will be recalled now [LS1], [La]. Let A = {a_{1} < a_{2} < … < a_{n}} denote a totally ordered alphabet of noncommutative indeterminates, and consider the free monoid A^{∗} generated by A. The Robinson-Schensted bijection associates with a word w ∈ A^{∗} a couple (P (w), Q(w)) of Young tableaux of the same shape. It was shown by Knuth [Kn] that the equivalence on A^{∗} defined by

w ≡ w′ ⟺ P (w) = P (w′)

is generated by the relations

zxy ≡ xzy , yxz ≡ yzx , yxx ≡ xyx , yxy ≡ yyx ,

for any x<y<z in A. The quotient set A^{∗}/≡, which by definition is in one to one correspondence with the set of Young tableaux on A, is therefore endowed with a multiplicative structure reflecting the Robinson-Schensted construction.

This multiplication being noncommutative, Lascoux and Schützenberger introduced the operation of cyclage, a monoid analogue of the conjugation h → g^{−1}hg in noncommutative groups [LS1], [La]. To be precise, given t, t′ in the plactic monoid A^{∗}/≡ and i ≥ 2, write

t i\overline{}t′

if and only if there exists u in A^{∗}/≡ such that t = a_{i} u and t′ = u a_{i}. In this way, Tab (·, μ) is given the structure of a connected oriented graph, whose transitive closure is a partial order with minimal element the row tableau t_{μ} := a_{1}^{μ_{1}}a_{2}^{μ_{2}} ⋯ a_{n}^{μ_{n}}. For example, if μ = (2, 2, 1), one obtains the ranked poset shown in Figure 1, in which for convenience the letters a_{i} in the tableaux have been replaced by their indices i. The cocharge co(t) of a tableau t is defined as its rank in the poset Tab (· , μ), that is, the number of cyclages needed to transform t into the row tableau t_{μ}. Denote by m(μ) the maximal value of the cocharge on Tab (· , μ) (if μ is a partition, m(μ) = n(μ) = ∑_{i}(i − 1)μ_{i}). The charge of t is c(t) = m(μ) − co(t) and one has

Theorem 7.1 The Kostka-Foulkes polynomial K_{λμ}(q) is the generating function of the charge on Tab (λ, μ):

K_{λμ}(q) = ∑_{t∈Tab (λ,μ)} q^{c(t)} .

[figure: 1 1 2 2 3; ^{1 1 2 2 3}; ^{1 1 2 2 3}; ^{1 1 2 3 2}; ^{1 1 3 2 2}; ^{1 1 2 32}; ^{1 1 2 2 3}; 3; 2; 2; 2; 3; 3; 2; 0; 1; 2; 3; 4; 4; 3; 2; 1; 0; Cocharge; Charge]

Figure 1:

Given a tableau t of shape λ, put s_{t} := s_{λ}. With this notation, the theorem can be stated as follows.

Corollary 7.2 The modified Hall-Littlewood function Q′_{μ} is equal to

Q′_{μ} = ∑_{t∈Tab (· , μ)} q^{c(t)} s_{t} .

Example 7.3 The description given in Figure 1 of the ranked poset Tab (. , (2, 2, 1)) yields the expansion

Q′_{221} = s_{221} + (q + q^{2})s_{32} + qs_{311} + (q^{2} + q^{3})s_{41} + q^{4}s_{5} .

Alternatively, one can compute the charge of a Young tableau t whose weight is a partition μ using the following algorithm.

Suppose first that μ = (1^{n}). Let w denote the word obtained by reading t from left to right and top to bottom. Attach an index to each letter of w as follows; the letter a_{1} has index 0, and if a_{r} has index i, then a_{r+1} has index i or i + 1 according as it lies to the left or right of a_{r}. The charge of t (or w) is then the sum of the indices.

Now if μ is any partition, the row-reading w of t is decomposed into a sequence of standard words w_{i}. To obtain w_{1}, read w from the right and choose the first letter a_{1}, then the first letter a_{2} to the left of the a_{1} chosen, and so on. If at any stage there is no a_{r+1} to the left of the a_{r} chosen, go back to the right end of w. This procedure extracts a standard subword w_{1} of w of length μ′_{1}. Now erase w_{1} from w and repeat the procedure to get a standard subword w_{2} of length μ′_{2}, and so on. The charge of t is the sum of the charges of the w_{i}.

Example 7.4 Take

\begin{matrix}1 & 1 & 2 & 3 & 3 & 2 & 4 & 5 & 4 & 5 & 6 & 6 \\ t = \end{matrix}

where for short the letter a_{r} is simply written r. One has w = 6 6 4 5 2 4 5 1 1 2 3 3 and w_{1} = 6 5 2 4 1 3, w_{2} = 6 4 5 1 2 3. Hence, c(t) = c(w_{1}) + c(w_{2}) = 4 + 11 = 15.

The combinatorial description of Hall-Littlewood functions and Kostka polynomials provides some new methods of investigation. Thus, it enabled Han to prove the following conjecture of Gupta for the growth of Kostka polynomials [Gu3], [Han].

Given a partition λ and a positive integer a, denote by λ ∪ a the partition obtained by inserting in λ a new part equal to a.

Theorem 7.5 For any partitions λ, μ and integer a, the difference

K_{λ∪a,μ∪a}(q) − K_{λμ}(q)

is a polynomial with nonnegative coefficients.

Han’s proof consists in the construction of an injective map from Tab (λ, μ) to Tab (λ∪ a, μ ∪ a) which preserves the charge of Young tableaux.

Another proof of Theorem 7.5 has been given by Kirillov [Ki4]. It is based on the theory of rigged configurations that we shall now present.

8 Kostka-Foulkes polynomials, Bethe Ansatz and configurations

A different description of Kostka-Foulkes polynomials was obtained by Kirillov and Reshetikhin [KR] as a consequence of their investigation of combinatorial aspects of the quantum inverse scattering method.

The state space of certain integrable quantum systems, called 𝔤𝔩 (n, C)-invariant models, is equal to the tensor product

𝔥 = S^{μ_{1}}(V) ⊗ ⋯ ⊗ S^{μ_{r}}(V)

of symmetric powers of the defining representation V of 𝔤𝔩 (n, C). The eigenvectors of the Hamiltonian of the model can be constructed by means of a substitution known as the Bethe Ansatz [Fa]. They are parametrized by the solutions of a system of algebraic equations (Bethe equations) which in turn are in one to one correspondence with combinatorial objects named rigged configurations [TF]. It can be shown analytically that the Bethe vectors form a complete system of highest weight vectors for the 𝔤𝔩 (n, C)-module 𝔥 and hence can also be parametrized by Young tableaux of weight μ = (μ_{1}, …, μ_{r}) [Ki1][Ki2]. In [KKR] [KR] an explicit bijection is constructed between Young tableaux and rigged configurations, which sends the charge of Young tableaux on a natural invariant of rigged configurations, namely, the sum of their quantum numbers. This yields a very efficient formula for evaluating Kostka-Foulkes polynomials as sums of products of q-binomial coefficients.

Before explaining the Kirillov-Reshetikhin bijection, we introduce the necessary definitions concerning configurations. Let λ, μ be partitions such that |λ| = |μ|. It is convenient to regard λ, μ as infinite nonincreasing sequences by putting λ_{k} = 0 for k > ℓ(λ), μ_{k} = 0 for k > ℓ(μ). A matrix of type (λ, μ) is a matrix M = (m_{ij})_{i,j≥1} with only a finite number of nonzero integer entries, such that

∑_{j≥1} m_{ij} = λ_{i} , ∑_{i≥1} m_{ij} = μ′_{j} .

Here and in the sequel, we denote by μ′_{j} the jth part of the conjugate partition μ′ of μ.

With such a matrix M are associated two matrices P = (P_{ij})_{i,j≥1}, Q = (Q_{ij})_{i,j≥1} defined by

P_{ij} = ∑_{k≤j}(m_{ik} − m_{i+1,k}) , Q_{ij} = ∑_{k≥i+1}(m_{kj} − m_{k,j+1}) .

The matrix M is an admissible matrix of type (λ, μ) if and only if P_{ij} and Q_{ij} are nonnegative for all i, j ≥ 1. The numbers P_{ij} such that Q_{ij} > 0 are called the vacancy numbers of M.

An admissible matrix is conveniently pictured as a sequence ν = (ν^{0}, ν^{1}, ν^{2}, …) of partitions, or configuration, defined by

(ν^{i})′_{j} = ∑_{k≥i+1} m_{kj} .

Obviously, one can recover the admissible matrix M from the configuration ν, so that the two combinatorial datas M or ν are equivalent. In particular, the type (λ, μ) is computed from ν by λ_{i} = |ν^{i}| − |ν^{i−1}|, for i ≥ 1, and μ = ν^{0}.

The rows of the diagram of ν^{i} are called strings of type i. By construction the nonzero integers Q_{ij} in the ith row of Q are equal to the multiplicities of the partition ν^{i}, i.e. to the successive widths of the blocks of strings of equal length in ν^{i}, so that a vacancy number P_{ij} is in fact associated with each block of strings of type i ≥ 1. It is written to the right of the block.

Example 8.1 Let us illustrate all these definitions. The matrix

M = ( \begin{matrix}3 & 5 & −2 & −1 \\ 2 & 0 & 1 & 1 \\ 1 & 1 & 1 & 0 \\ 1 & 1 & 0 & 0 \end{matrix} )

has type (λ, μ), where λ = (5, 4, 3, 2) and μ = (2, 2, 2, 2, 2, 2, 2). The associated matrices P and Q are equal to

P = ( \begin{matrix}1 & 6 & 3 & 1 \\ 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 1 \end{matrix} ) , Q = ( \begin{matrix}2 & 0 & 1 & 1 \\ 0 & 1 & 1 & 0 \\ 0 & 1 & 0 & 0 \end{matrix} ) .

The nonzero entries of Q and the corresponding vacancy numbers in P have been printed in bold type.

The representation of M as a configuration ν = (ν^{0}, ν^{1}, ν^{2}, ν^{3}) is

1 3 1  ^{0 0}  0

The last definition is that of a rigged configuration, by which we mean a configuration ν together with a family of quantum numbers. These are nonnegative integers I_{j}^{i} attached to each string ν_{j}^{i} of type i ≥ 1, which must not exceed the vacancy number associated with their block. Quantum numbers are written in the leftmost cell of the string to which they belong. The order of the quantum numbers inside a given block is irrelevant. In other words, permutations of the strings of a same block are allowed, so that we can always assume that quantum numbers are weakly increasing (or decreasing) from bottom to top within each block.

Example 8.2 The following is a rigging of the configuration ν of Example 8.1.

0301  1 3 1 ^{00  0 0} 0 0

It is clear from the definition that the number of rigged configurations (ν, I) corresponding to a given configuration ν is equal to the product of binomial coefficients

N = ∏_{i,j≥1} \binom{P_{ij} + Q_{ij}}{Q_{ij}}

(with the usual convention \binom{m}{0} = 1 for m ∈ Z). As stated above, the Bethe vectors of 𝔥 are highest weight vectors parametrized by rigged configurations. More precisely, the Bethe vectors of weight λ in S^{μ_{1}}(V) ⊗ ⋯ ⊗ S^{μ_{r}}(V) are labelled by rigged configurations of type (λ, μ). It follows from the completeness of the Bethe system that Theorem 8.3 The Kostka number K_{λμ} is equal to

K_{λμ} = ∑_{ν} N(ν) = ∑_{ν} ∏_{i,j≥1} \binom{P_{ij}(ν) + Q_{ij}(ν)}{Q_{ij}(ν)}

the sum running over all configurations ν of type (λ, μ).

Example 8.4 Take λ = (8, 5, 2), μ = (6, 4, 3, 2). There are two configurations ν of type (λ, μ), viz.

2 2  0

1 1  0

Thus K_{λμ} = \binom{2+1}{1}\binom{2+1}{1}\binom{0+1}{1} + \binom{1+1}{1}\binom{1+1}{1}\binom{0+1}{1} = 9 + 4 = 13.

We shall now explain the bijection devised by Kirillov and Reshetikhin for proving combinatorially Theorem 8.3.

A string ν_{j}^{i} of type i ≥ 1 in a rigged configuration (ν, I) is called singular when its quantum number I_{j}^{i} is maximal, i.e. is equal to the corresponding vacancy number. By convention, the only singular string of type 0 is the upper one. Consider now chains of weakly increasing singular strings

ν_{i_{0}}^{0} ≤ ν_{i_{1}}^{1} ≤ ⋯ ≤ ν_{i_{s}}^{s}

in (ν, I). Define the rank of (ν, I) as the maximal length of such a chain. For example, the rank of the rigged configuration shown in Example 8.2 is equal to 3, the longest chain being

3  3  0  0

Let t be a Young tableau of shape λ and weight μ. The cells of t can be ordered by first reading the 1’s from left to right, then the 2’s from left to right, and so on (this is the usual standardization of t). The rigged configuration (ν, I) corresponding to t is now constructed inductively as follows.

The last cell of t belongs to row number r, say. Let \overline{t} be the tableau obtained from t by erasing this cell, and assume that the corresponding rigged configuration (\overline{ν}, \overline{I}) is known.

Construction of ν: Add one cell to one of the longest singular strings of type r − 1 in (\overline{ν}, \overline{I}) (if there is none, create a new string of type r − 1 and length 1). Let m be the length of this new string. Similarly, add one cell to one of the longest singular strings of type r − 2 and length ≤ m − 1 (if there is none, create a new string of type r − 2 and length 1). And so on. If necessary, reorder the strings inside each diagram in order to get a sequence of partitions. In this way a new configuration ν of type (λ, μ) is obtained, which differs from \overline{ν} by a single chain C composed of r strings.

Construction of I: Compute the vacancy numbers of ν. The r strings composing C are given their maximal quantum numbers, while the quantum numbers of all the other strings remain unchanged.

Example 8.5 We take

\begin{matrix}1 & 1 & 2 & 3 & 3 & 2 & 4 & 5 & 4 & 5 & 6 & 6 \\ t = \end{matrix}

The configuration (\overline{ν}, \overline{I}) associated to

\begin{matrix}1 & 1 & 2 & 3 & 3 & 2 & 4 & 5 & 4 & 5 & 6 \\ \overline{t} = \end{matrix}

is supposed to be known, and is equal to

2000  2 0  ^{01 0 1}  0 0

Here, r = 4 and the rigged configuration (ν, I) computed by application of the rules described above is

2400  2 4 0  ^{00 0}  0 0

the associated matrices being

M = ( \begin{matrix}2 & 4 & −1 \\ 2 & 0 & 1 \\ 1 & 1 & 0 \\ 1 & 1 & 0 \end{matrix} ) , P = ( \begin{matrix}0 & 4 & 2 \\ 1 & 0 & 1 \\ 0 & 0 & 0 \end{matrix} ) , Q = ( \begin{matrix}2 & 1 & 1 \\ 0 & 2 & 0 \\ 0 & 1 & 0 \end{matrix} ) .

Example 8.6 We give now the complete construction of the rigged configuration associated to the tableau t of Example 8.5. One passes from one step to the following by application of the previous rules. For the sake of brevity, the matrices M, P, Q necessary at each step for the determination of the vacancy and quantum numbers have not been reproduced.

1

1 1

1 1 2  0 0

1 1 2 2  0 0

1 1 2 3 2  0 1

1 1 2 3 3 2  0 1

1 1 2 3 3 24  ^{01 1}  0 0

1 1 2 3 3 2 4 4  ^{30 3 1}  0 0

1 1 2 3 3 2 4 4 5  ^{300 3 0}  0 0

1 1 2 3 3 2 4 5 4 5  ^{200  2 0} 0 0

1 1 2 3 3 2 4 5 4 5 6  ^{2000  2 0} ^{01 0 1} 0 0

1 1 2 3 3 2 4 5 4 5 6 6  ^{2400  2 4 0} ^{00 0} 0 0

The most important property of the Kirillov-Reshetikhin bijection is that the charge of a Young tableau t can be computed very easily from the rigged configuration (ν, I) to which it corresponds.

Given a configuration ν with admissible matrix M = (m_{ij}), define the charge of ν (or M) by

c(ν) = c(M) = \frac{1}{2} ∑_{i,j} m_{ij}(m_{ij} − 1) .

Then one can show that

c(t) = c(ν) + ∑_{i,j} I_{j}^{i} ,

which yields the following q-analogue of 8.3

Theorem 8.7 The Kostka polynomial K_{λμ}(q) is equal to

K_{λμ}(q) = ∑_{ν} ∏_{i,j≥1} q^{c(ν)} [ \begin{matrix}P_{ij}(ν) + Q_{ij}(ν) \\ Q_{ij}(ν) \end{matrix} ]_{q} ,

the sum running over all configurations ν of type (λ, μ).

Example 8.8 Take λ = (8, 5, 2), μ = (6, 4, 3, 2) as in Example 8.4. The admissible matrices corresponding to the two configurations given in this example are respectively

( \begin{matrix}2 & 2 & 2 & 1 & 0 & 1 \\ 1 & 1 & 1 & 1 & 1 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 \end{matrix} ) ( \begin{matrix}2 & 2 & 1 & 1 & 1 & 1 \\ 1 & 1 & 2 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 \end{matrix} ) .

Their charges are both equal to 3, hence

K_{λμ}(q) = q^{3} [ \begin{matrix}3 \\ 1 \end{matrix} ]_{q} [ \begin{matrix}3 \\ 1 \end{matrix} ]_{q} + q^{3} [ \begin{matrix}2 \\ 1 \end{matrix} ]_{q} [ \begin{matrix}2 \\ 1 \end{matrix} ]_{q} = 2q^{3} + 4q^{4} + 4q^{5} + 2q^{6} + q^{7} .

An example of application of Theorem 8.7 is the proof by Kirillov of the unimodality of generalized Gaussian coefficients [Ki5]. In fact, when λ consists of only two parts, 8.7 gives a generalization of the KOH formula for q-binomial coefficients.

9 Specializations at roots of unity

When q = −1, the Hall-Littlewood functions coincide with those introduced by Schur in his theory of projective representations of the symmetric group.

The investigation of Hall-Littlewood functions at other roots of unity has been initiated by Morris [Mo2], in connection with problems in the modular representation theory of the symmetric group. Let p be a prime, ζ a primitive pth root of 1, and μ a partition with at least one multiplicity m_{i}(μ) ≥ p. It follows from (16) that Q_{μ}(X; ζ) = 0. On the other hand,

Q_{μ}(X; q) = ∑_{λ} K_{λ,μ}(q) s_{λ}((1 − q)X) = ∑_{λ,ρ} K_{λ,μ}(q) \frac{χ_{ρ}^{λ}}{z_{ρ}} p_{ρ}((1 − q)X)

= ∑_{ρ} (∑_{λ} K_{λ,μ}(q) χ_{ρ}^{λ}) \frac{∏_{i}(1 − q^{ρ_{i}})}{z_{ρ}} p_{ρ}(X) .

Hence, for all partitions ρ such that ρ_{i} ≢ 0 mod p, one has

∑_{λ} K_{λ,μ}(ζ) χ_{ρ}^{λ} = 0 .  (35)

Denote by π^{p}(n) the set of partitions ρ of n whose parts are not divisible by p, and by π_{p}(n) the set of partitions μ whose multiplicities are less than p. It is well known that p^{∗}(n) := |π^{p}(n)| = |π_{p}(n)|. The elements of π^{p}(n) correspond to p-regular conjugacy classes in S_{n}, that is, classes whose elements have an order prime to p, while π_{p}(n) parametrizes the pmodular irreducibles characters. A basic problem in the p-modular representation theory of S_{n} is to find a complete set of linearly independent relations between the characters χ^{λ} restricted to p-regular classes ρ. The number of such independent relations being equal to p(n) − p^{∗}(n), the system of identities (35) provides a solution to this question. Thus, in the case n = 4, p = 2, one obtains the three relations

χ_{ρ}^{1^{4}} − χ_{ρ}^{211} + 2χ_{ρ}^{22} − χ_{ρ}^{31} + χ_{ρ}^{4} = 0 , χ_{ρ}^{211} − χ_{ρ}^{22} − χ_{ρ}^{4} = 0 , χ_{ρ}^{22} − χ_{ρ}^{31} + χ_{ρ}^{4} = 0 ,

for ρ = (1^{4}), (3, 1).

The problem of specializing Hall-Littlewood functions at roots of unity is also related to some combinatorial properties of the so-called cyclic characters introduced by Foulkes [Fo2]. These symmetric functions correspond to characters of the symmetric group induced by irreducible representations of cyclic subgroups generated by a full cycle. A combinatorial interpretation of their coefficients in the basis of Schur functions has been given by Kraskiewicz and Weyman [KW] in terms of congruence properties of the major index of permutations, or, which amounts to the same, charge of standard tableaux. A generalization of this result has been proposed in [LLT2], in which the coefficients of certain plethysms with the cyclic characters are interpreted in terms of the charge of more general tableaux. This result can also be viewed as an isomorphism theorem for certain submodules of the cohomology of a generalized flag manifold, or also in terms of harmonic polynomials relative to the symmetric group.

Finally, these theorems can be interpreted in terms of congruence properties of Kostka-Foulkes polynomials modulo cyclotomic polynomials.

We shall review in this Section the main results of [LLT1] and [LLT2].

The first one is a factorization property of the functions Q′_{λ}(X, q) when q is specialized to a primitive root of unity. This is to be seen as a generalization of the fact that when q is specialized to 1 the function Q′_{λ}(X; q) reduces to h_{λ}(X) = ∏_{i} h_{λ_{i}}(X).

Theorem 9.1 Let λ = (1^{m_{1}}2^{m_{2}} … n^{m_{n}}) be a partition written multiplicatively. Set m_{i} = kq_{i} + r_{i} with 0 ≤ r_{i} < k, and μ = (1^{r_{1}}2^{r_{2}} … n^{r_{n}}). Then, ζ being a primitive k-th root of unity,

Q′_{λ}(X; ζ) = Q′_{μ}(X; ζ) ∏_{i≥1}[Q′_{(i^{k})}(X; ζ)]^{q_{i}} .  (36)

The functions Q′_{(i^{k})}(X; ζ) appearing in the right-hand side of (36) can be given a more convenient expression.

Theorem 9.2 Let p_{k} ∘ h_{n} denote the plethysm of the complete function h_{n} by the powersum p_{k}, which is defined by the generating series ∑_{n} p_{k}∘h_{n}(X) z^{n} = ∏_{x∈X}(1−zx^{k})^{−1}. Then,

if ζ is as above a primitive k-th root of unity, one has

Q′_{(n^{k})}(X; ζ) = (−1)^{(k−1)n}p_{k} ∘ h_{n}(X).

Example 9.3 With k = 3 (ζ = e^{2iπ/3}), we have

Q′_{444433311}(X; ζ) = Q′_{411}(X; ζ) Q′_{333}(X; ζ) Q′_{444}(X; ζ) = Q′_{411}(X; ζ) p_{4} ∘ h_{43} .

One can derive from 9.1, 9.2 the following results for Green polynomials at roots of unity, which had been conjectured by Morris and Sultana [MS] [Su]:

Theorem 9.4 Let λ = (1^{m_{1}}2^{m_{2}} … n^{m_{n}}) be a partition, with at least one part i with multiplicity m_{i} ≥ k, and let μ = (μ_{1}, …, μ_{r}) be another partition, with at least one part μ_{h} ≡ 0 (mod k). Then,

X_{μ}^{λ}(ζ) ≡ 0 (mod k) .

Theorem 9.5 Let λ = (1^{m_{1}} … i^{m_{i}} … n^{m_{n}}), where for some i, m_{i} ≥ 1, |λ| = n, λ^{∗} = (1^{m_{1}} … i^{m_{i}−1} … n^{m_{n}}) and suppose n = kr. Then,

X_{λ}^{(r^{k})}(ζ) = (−1)^{(k−1)j}k X_{λ^{∗}}^{(r−j)^{k}}(ζ)

if i = jk, and X_{λ}^{(r^{k})}(ζ) = 0 otherwise.

Example 9.6 With k = 4 (ζ = i = \sqrt{−1}), we have

X_{422}^{41111}(i) = ⟨Q′_{1111}Q′_{4} , p_{4}p_{22}⟩ = (−1)^{3}⟨p_{4}Q′_{4} , p_{4}p_{22}⟩
= −⟨Q′_{4} , D_{p_{4}}(p_{4}p_{22})⟩ = −⟨Q′_{4} , 4p_{22}⟩ = −4⟨h_{4} , p_{22}⟩ = −4 .

Theorem 9.5 may in fact be generalized as follows.

Given two partitions λ and μ, we denote by λ ∨ μ the partition obtained by reordering the concatenation of λ and ν, e.g. (2, 2, 1) ∨ (5, 2, 1, 1) = (5, 2^{3}, 1^{3}). We write μ^{k} = μ ∨ μ ∨ ⋯ ∨ μ (k factors). If μ = (μ_{1}, …, μ_{r}), we set kμ = (kμ_{1}, …, kμ_{r}). Let λ = μ^{k}, and let ν be another partition. The Green polynomial X_{ν}^{λ}(ζ) can then be expressed by means of the value of a permutation character of S_{n}:

X_{ν}^{μ^{k}}(ζ) = (−1)^{(k−1)|μ|}⟨p_{ν}, p_{k}(h_{μ})⟩ = (−1)^{(k−1)|μ|}⟨ϕ_{k}(p_{ν}), h_{μ}⟩ ,

where ϕ_{k} is the adjoint of the linear operator ψ^{k} : F ↦ p_{k} ∘ F (Adams operator), so that:

Theorem 9.7 X_{ν}^{μ^{k}}(ζ) = 0 when ν is not of the form kπ for some partition π, and

X_{kπ}^{μ^{k}}(ζ) = (−1)^{(k−1)|μ|}k^{ℓ(π)}⟨p_{π} , h_{μ}⟩ .

In particular, X_{kπ}^{(r^{k})}(ζ) = (−1)^{(k−1)r}k^{ℓ(π)}.

Example 9.8 (i) With k = 4 (ζ = i), X_{422}^{2222}(i) = ⟨p_{4}(h_{2}), p_{422}⟩ = 0 and

X_{44}^{42}(i) = ⟨h_{2}, ϕ_{4}(p_{44})⟩ = ⟨h_{2}, 16p_{11}⟩ = 16 .

(ii) With k = 3 (ζ = e^{2πi/3}),

X_{933}^{333222}(ζ) = +3^{3}⟨p_{311}, h_{23}⟩ = 3^{3}⟨p_{11}, h_{2}D_{p_{3}}h_{3}⟩ = 27 .

To state our next result, we recall some definitions from number theory. For k, n ∈ N, the Ramanujan (or Von Sterneck) sum c(k, n) (also denoted Φ(k, n)) is the sum of the k-th powers of the primitive n-th roots of unity. Its value is given by Hölder’s formula: if (k, n) = d and n = md, then c(k, n) = μ(m)ϕ(n)/ϕ(m), where μ is the Moebius function and ϕ is the Euler totient function (see e.g. [HW] or [NV]).

Let P (q) = ∑_{k=0}^{n−1} a_{k} q^{k} ∈ Z[q] be a polynomial of degree ≤ n − 1. P is said to be even modulo n if (i, n) = (j, n) ⇒ a_{i} = a_{j}. The following result, due to E. Cohen ([Co], see also [De1] for a simpler proof), provides a generalization of the Moebius inversion formula:

Lemma 9.9 The polynomial P is even modulo n iff for every divisor d of n, the residue of P (q) modulo the cyclotomic polynomial Φ_{d}(q) is a constant r_{d} ∈ Z. In this case, one has

a_{k} = \frac{1}{n} ∑_{d|n} c(k, d)r_{d} , r_{d} = ∑_{t|n} c(n/d, t)a_{n/t} .

With the aid of Ramanujan sums, we define the symmetric functions

ℓ_{n}^{(k)} = \frac{1}{n} ∑_{d|n} c(k, d)p_{d}^{n/d} .  (37)

These functions were first obtained by Foulkes as Frobenius characteristics of the representations of the symmetric group induced by irreducible representations of a cyclic subgroup generated by an n-cycle [Fo2]. A combinatorial interpretation of the multiplicity ⟨s_{λ} , ℓ_{n}^{(k)}⟩ has been given by Kraskiewicz and Weyman [KW]. This result is equivalent to the congruence

Q′_{1^{n}}(X; q) ≡ ∑_{0≤k≤n−1} q^{k}ℓ_{n}^{(k)} (mod 1 − q^{n}) .

A proof using Cohen’s formula can be found in [De1]. Adapting this argument to our case, we arrive at the following generalization of the result of [KW]:

Theorem 9.10 Let e_{i} be the i-th elementary symmetric function, and for λ = (λ_{1}, …, λ_{m}), e_{λ} = e_{λ_{1}} ⋯ e_{λ_{r}}. Then, the multiplicity ⟨s_{μ} , ℓ_{k}^{(r)} ∘ e_{λ}⟩ of the Schur function s_{μ} in the plethysm ℓ_{k}^{(r)} ∘ e_{λ} is equal to the number of Young tableaux of shape μ′ (conjugate partition) and weight λ^{k} whose charge is congruent to r modulo k.

Example 9.11 With k = 4, r = 2 and λ = (2),

ℓ_{4}^{(2)} ∘ e_{2} = s_{431} + s_{422} + s_{41111} + 2s_{3311} + 2s_{3221}
+2s_{32111} + s_{2222} + s_{22211} + 2s_{221111} + s_{2111111} .

To compute the coefficient ⟨s_{32111} , ℓ_{4}^{(2)} ∘ e_{2}⟩ = 2, we have to find the number of tableaux of shape (3, 2, 1, 1, 1)′ = (5, 2, 1), weight (2, 2, 2, 2) and charge ≡ 2 (mod 4). The two tableaux satisfying these constraints are:

1 1 2 3 4 2 4 3  ^{1 1 2 3 4 2 3 4}

which both have charge equal to 6.

Similarly, the reader can check that ⟨s_{732} , ℓ_{4}^{(2)} ∘ e_{21}⟩ = 5 is the number of tableaux with shape (3, 3, 2, 1, 1, 1, 1), weight (2, 2, 2, 2, 1, 1, 1, 1) and charge ≡ 2 (mod 4).

A related result (see e.g. [MT]), gives an interpretation of the cyclic characters ℓ_{n}^{(k)} in terms of S_{n}-harmonic polynomials: let H^{k} be the Frobenius characteristic of the homogeneous component H^{k} of degree k of the module H of S_{n}-harmonic polynomials. Then,

ℓ_{n}^{(k)} = ∑_{i≡k (mod n)} H^{i} .

More generally, the plethysm ℓ_{k}^{(j)} ∘ e_{λ} can be expressed as a positive sum of Frobenius characteristics of modules of harmonic polynomials. Recall that to each partition μ of n, it is possible to associate a certain submodule L_{μ} of H_{n} with graded characteristic

F_{q}(L_{μ}) = Q̃′_{μ}(X; q) = ∑_{k} q^{k}H_{μ}^{k}(X) .  (38)

Theorem 9.12 Let μ = λ^{k}. Then,

ℓ_{k}^{(j)} ∘ e_{λ} = ∑_{i≡j (mod k)} H_{μ}^{i} .

Example 9.13 The character of the preceding example decomposes into three parts

ℓ_{4}^{(2)} ∘ e_{2} = H_{2222}^{2} + H_{2222}^{6} + H_{2222}^{10},

the three components being

H_{2222}^{2} = s_{221111} + s_{2111111} ,
H_{2222}^{6} = s_{3311} + 2s_{3221} + s_{2222} + s_{41111} + 2s_{32111} + s_{22211} + s_{221111} ,
H_{2222}^{10} = s_{431} + s_{422} + s_{3311} .

Theorems 9.1, 9.2 may also be stated in terms of specialization of Kostka-Foulkes polynomials at roots of unity, or equivalently, in terms of congruence properties of the K_{λ,μ}(q) modulo cyclotomic polynomials.

Theorem 9.14 For any partition λ with |λ| = kn,

K_{λ,n^{k}}(q) ≡ (−1)^{(k−1)n} s_{λ}(1, q, q^{2}, …, q^{k−1}) (mod Φ_{k}(q))

and more generally, for any partition μ with |μ| = n,

K_{λ,μ^{k}}(q) ≡ (−1)^{(k−1)|μ|} (h_{kμ} ∗ s_{λ})(1, q, q^{2}, …, q^{k−1}) (mod Φ_{k}(q)) .

We mention that a different type of congruence follows from results of Stanley [Sta] and Gupta [Gu1], that is, for β_{1} = n and

λ = [α, β]_{k} := (α_{1} + n, …, α_{s} + n, n, …, n, n − β_{t}, n − β_{t−1}, …, n − β_{2}, 0)

then (cf. [Ki4])

K_{λ,(n^{k})}(q) ≡ (s_{α} ∗ s_{β})(q, …, q^{k−1}) (mod q^{k}) .

Theorem 9.14 gives some congruences modulo cyclotomic polynomials for certain columns of the q-Kostka matrix. We can also derive from 9.1, 9.2 other congruences for certain rows of the same matrix.

Let ρ_{n} be the staircase partition (n, …, 2, 1). It is well-known that for k = 2 (ζ = −1), one has s_{ρ_{n}}(X) = P_{ρ_{n}}(X; −1), the staircase P -Schur function. More generally, it can be shown that if ζ is a primitive k-th root of unity, P_{ρ_{n}^{k−1}}(X; ζ) = s_{ρ_{n}^{k−1}}(X). This is a consequence of the following result:

Theorem 9.15 Let ζ be a primitive k-th root of 1, and λ be a partition such that ℓ(λ′) ≤ n. Then,

s_{λ^{k}∨ρ_{n}^{k−1}}(X) = ∑_{μ} K_{λμ} P_{μ^{k}∨ρ_{n}^{k−1}}(X; ζ)

where K_{λμ} are the usual Kostka numbers. In other words, we have the following congruence modulo the cyclotomic polynomial Φ_{k}(q):

K_{λ^{k}∨ρ_{n}^{k−1}, ν}(q) ≡ \{ \begin{matrix}K_{λμ} & if & ν = μ^{k} ∨ ρ_{n}^{k−1} \\ 0 & otherwise. \end{matrix}

T^{1} = ^{1 2 3 4  5 6  7 8  9}  T^{2} = ^{4 2 6 1  7 8  5 3  9} Figure 2:

Example 9.16 Take k = 3, n = 3, I = (3, 2) and J = (2, 2, 1). Then,

K_{333332222211,332222222211111}(q) = q^{3}(1 + 2q + 6q^{2} + 13q^{3} + 24q^{4} + 39q^{5}

+59q^{6} +81q^{7} +105q^{8} +128q^{9} +148q^{10} +163q^{11} +171q^{12} +172q^{13} +165q^{14} +153q^{15} +134q^{16}

+114q^{17} + 92q^{18} + 72q^{19} + 53q^{20} + 38q^{21} + 25q^{22} + 16q^{23} + 9q^{24} + 5q^{25} + 2q^{26} + q^{27})

≡ K_{32,221} = 2 (mod 1 + q + q^{2}) .

An alternative and more combinatorial formulation of Theorems 9.1 and 9.2 may be presented by introducing the notion of ribbon tableau.

Recall that to a partition λ is associated a k-core λ_{(k)} and a k-quotient λ^{(k)} [JK]. The k-core is the unique partition obtained by successively removing k-ribbons (or skew hooks) from λ. The different possible ways of doing so can be distinguished from one another by labelling 1 the last ribbon removed, 2 the penultimate, and so on. Thus Figure 2 shows two different ways of reaching the 3-core λ_{(3)} = (2, 1^{2}) of λ = (8, 7^{2}, 4, 1^{5}). These pictures represent two 3-ribbon tableaux T_{1}, T_{2} of shape λ/λ_{(3)} and weight μ = (1^{9}).

To define k-ribbon tableaux of general weight and shape, we need some terminology. The initial cell of a k-ribbon R is its rightmost and bottommost cell. Let θ = β/α be a skew shape, and set α_{+} = (β_{1}) ∨ α, so that α_{+}/α is the horizontal strip made of the bottom cells of the columns of θ. We say that θ is a horizontal k-ribbon strip of weight m, if it can be tiled by m k-ribbons the initial cells of which lie in α_{+}/α. (One can check that if such a tiling exists, it is unique).

Now, a k-ribbon tableau T of shape λ/ν and weight μ = (μ_{1}, …, μ_{r}) is defined as a chain of partitions

ν = α^{0} ⊂ α^{1} ⊂ ⋯ ⊂ α^{r} = λ

such that α^{i}/α^{i−1} is a horizontal k-ribbon strip of weight μ_{i}. Graphically, T may be described by numbering each k-ribbon of α^{i}/α^{i−1} with the number i. We denote by Tab_{k}(λ/ν, μ) the set of k-ribbon tableaux of shape λ/ν and weight μ, and we set

K_{λ/ν, μ}^{(k)} = |Tab_{k}(λ/ν, μ)| .

Finally we recall the definition of the k-sign ϵ_{k}(λ/ν). Define the sign of a ribbon R as (−1)^{h−1}, where h is the height of R. The k-sign ϵ_{k}(λ/ν) is the product of the signs of all the ribbons of a k-ribbon tableau of shape λ/ν (this does not depend on the particular tableau chosen, but only on the shape).

The origin of these combinatorial definitions is best understood by analyzing carefully the operation of multiplying a Schur function s_{ν} by a plethysm of the form p_{k} ∘ h_{μ}. Equivalently, thanks to the involution ω, one may rather consider a product of the type s_{ν} [p_{k} ∘ e_{μ}]. To this end, since

p_{k} ∘ e_{μ} = (e_{μ_{1}} ∘ p_{k}) ⋯ (e_{μ_{n}} ∘ p_{k}) = m_{k^{μ_{1}}} ⋯ m_{k^{μn}}

one needs only to apply repeatedly the following multiplication rule due to Muir [Mu] (see also [Li3]):

s_{ν} m_{α} = ∑_{β} s_{ν+β} ,

sum over all distinct permutations β of (α_{1}, α_{2}, …, α_{n}, 0, …). Here, as in the righthand side of (6), the Schur functions s_{ν+β} are not necessary indexed by partitions and have therefore to be standardized, this reduction yielding only a finite number of nonzero summands. For example,

s_{31} m_{3} = s_{61} + s_{313} + s_{31003} = s_{61} − s_{322} + s_{314} .

Other terms such as s_{34} or s_{3103} reduce to 0. It is easy to deduce from this rule that the multiplicity

⟨s_{ν} m_{k^{μ_{i}}} , s_{λ}⟩

is nonzero iff λ′/μ′ is a horizontal k-ribbon strip of weight μ_{i}, in which case it is equal to ϵ_{k}(λ/ν). Hence, applying ω we arrive at the expansion

s_{ν} [p_{k} ∘ h_{μ}] = ∑_{λ} ϵ_{k}(λ/μ) K_{λ/ν ,μ}^{(k)} s_{λ}

from which we deduce by 9.1, 9.2 that

K_{λ μ}^{(k)} = (−1)^{(k−1)|μ|} ϵ_{k}(λ) K_{λ μ^{k}}(ζ)

and more generally, defining as in [KR] the skew Kostka-Foulkes polynomial K_{λ/ν ,α}(q) by

K_{λ/ν ,α}(q) = ⟨s_{λ/ν} , Q′_{α}(q)⟩

we can write

K_{λ/ν ,μ}^{(k)} = (−1)^{(k−1)|μ|} ϵ_{k}(λ/ν) K_{λ/ν ,μ^{k}}(ζ) .

It turns out that enumerating k-ribbon tableaux is equivalent to enumerating k-uples of ordinary Young tableaux, as shown by the correspondence to be described now. This bijection was first studied by Stanton and White [SW] in the case of ribbon tableaux of right shape λ (without k-core) and standard weight μ = (1^{n}) (see also [FS]). We need some additional definitions.

Let R be a k-ribbon of a k-ribbon tableau. R contains a unique cell with coordinates (x, y) such that y −x ≡ 0 (mod k). We decide to write in this cell the number attached to R, and we define the type i ∈ {0, 1, …, k − 1} of R as the distance between this cell and the initial cell of R. For example, the 3-ribbons of T_{1} are divided up into three classes:

• 4, 6, 8, of type 0;

• 1, 2, 7, 9, of type 1;

• 3, 5, of type 2.

Define the diagonals of a k-ribbon tableau as the sequences of integers read along the straight lines D_{i} : y − x = ki. Thus T_{1} has the sequence of diagonals

((8), (4), (2, 3, 6), (1, 5, 9), (7)) .

This definition applies in particular to 1-ribbon tableaux, i.e. ordinary Young tableaux. It is obvious that a Young tableau is uniquely determined by its sequence of diagonals. Hence, we can associate to a given k-ribbon tableau T of shape λ/ν a k-uple (t_{0}, t_{1}, …, t_{k−1}) of Young tableaux defined as follows; the diagonals of t_{i} are obtained by erasing in the diagonals of T the labels of all the ribbons of type ≠ i. For instance, if T = T_{1} the first ribbon tableau of Figure 2, the sequence of diagonals of t_{1} is ((2), (1, 9), (7)), and

\begin{matrix}1 & 7 & 2 & 9 \\ t_{1} = \end{matrix}

The complete triple (t_{0}, t_{1}, t_{2}) of Young tableaux associated to T_{1} is

τ^{1} = ( ^{4 6 8} , ^{1 7 2 9} , 3 5 )

whereas that corresponding to T_{2} is

τ^{2} = ( ^{1 8 3} , ^{4 5 6 9} , 2 7 )

One can show that if ν = λ_{(k)}, the k-core of λ, the k-uple of shapes (λ^{0}, λ^{1}, …, λ^{k−1}) of (t_{0}, t_{1}, …, t_{k−1}) depends only on the shape λ of T, and is equal to the k-quotient λ^{(k)} of λ. Moreover the correspondence T ⟶ (t_{0}, t_{1}, …, t_{k−1}) establishes a bijection between the set of k-ribbon tableaux of shape λ/λ_{(k)} and weight μ, and the set of k-uples of Young tableaux of shapes (λ^{0}, …, λ^{k−1}) and weights (μ^{0}, …, μ^{k−1}) with μ_{i} = ∑_{j} μ_{i}^{j}. (See [SW] or [FS] for a proof in the case when λ_{(k)} = (0) and μ = (1^{n})).

For example, keeping λ = (8, 7^{2}, 4, 1^{5}), the triple

τ = ( ^{3 3 4} , ^{1 3 2 4} , 2 3 )

with weights ((0, 0, 2, 1), (1, 1, 1, 1), (0, 1, 1, 0)) corresponds to the 3-ribbon tableau

T = ^{1 2 2 3  3 3  3 4  4}

of weight μ = (1, 2, 4, 2).

As before, the significance of this combinatorial construction becomes clearer once interpreted in terms of symmetric functions. Recall the definition of ϕ_{k}, the adjoint of the linear operator ψ^{k} : F ↦ p_{k} ∘ F acting on the space of symmetric functions. In other words, ϕ_{k} is characterized by

⟨ϕ_{k}(F) , G⟩ = ⟨F , p_{k} ∘ G⟩ , F, G ∈ Sym .

Littlewood has shown [Li3] that if λ is a partition whose k-core λ_{(k)} is null, then

ϕ_{k}(s_{λ}) = ϵ_{k}(λ) s_{λ^{0}} s_{λ^{1}} ⋯ s_{λ^{k−1}}  (39)

where λ^{(k)} = (λ^{0}, …, λ^{k−1}) is the k-quotient. Therefore,

K_{λ μ}^{(k)} = ϵ_{k}(λ) ⟨p_{k} ∘ h_{μ} , s_{λ}⟩ = ϵ_{k}(λ) ⟨ϕ_{k}(s_{λ}) , h_{μ}⟩ = ⟨s_{λ^{0}} s_{λ^{1}} ⋯ s_{λ^{k−1}} , h_{μ}⟩

is the multiplicity of the weight μ in the product of Schur functions s_{λ^{0}} ⋯ s_{λ^{k−1}}, that is, is equal to the number of k-uples of Young tableaux of shapes (λ^{0}, …, λ^{k−1}) and weights (μ^{0}, …, μ^{k−1}) with μ_{i} = ∑_{j} μ_{i}^{j}. Thus, the bijection described above gives a combinatorial proof of (39).

More generally, if λ is replaced by a skew partition λ/ν, (39) becomes [KSW]

ϕ_{k}(s_{λ/ν}) = ϵ_{k}(λ/ν) s_{λ^{0}/ν^{0}} s_{λ^{1}/ν^{1}} ⋯ s_{λ^{k−1}/ν^{k−1}}

if λ_{(k)} = ν_{(k)}, and 0 otherwise. This can also be deduced from the previous combinatorial correspondence, but we shall not go into further details.

Returning to Kostka polynomials, we may summarize this discussion by stating Theorems 9.1 and 9.2 in the following way:

Theorem 9.17 Let λ and ν be partitions and set ν = μ^{k} ∨ α with m_{i}(α) < k. Denoting by ζ a primitive kth root of unity, one has

K_{λ, ν}(ζ) = (−1)^{(k−1)|μ|} ∑_{β} ϵ_{k}(λ/β) K_{λ/β, μ}^{(k)} K_{β, α}(ζ) .  (40)

Example 9.18 We take λ = (4^{2}, 3), ν = (2^{2}, 1^{7}) and k = 3 (ζ = e^{2iπ/3}). In this case, ν = μ^{k} ∨α with μ = (1^{2}) and α = (2^{2}, 1). The summands of (40) are parametrized by the 3-ribbon tableaux of external shape λ and weight μ. Here we have three such tableaux:

1 2  ^{1 2}  ^{1 2}

so that

K_{443, 221111111}(ζ) = 2K_{41, 221}(ζ) − K_{32, 221}(ζ) = 2(ζ^{2} + ζ^{3}) − (ζ + ζ^{2}) = 2ζ^{2} + 3 .

When |α| ≤ |λ_{(k)}|, (40) becomes simpler. For if |α| < |λ_{(k)}| then K_{λ, ν}(ζ) = 0, and otherwise the sum reduces to one single term

K_{λ, ν}(ζ) = (−1)^{(k−1)|μ|}ϵ_{k}(λ/λ_{(k)}) K_{λ/λ_{(k)}, μ}^{(k)} K_{λ_{(k)}, α}(ζ) .

In particular, if ν = (1^{n}), one recovers the following theorem of Morris and Sultana [MS].

Theorem 9.19 Let λ be a partition of n and ζ a primitive kth root of unity. Denote by H(λ^{(k)}) the product of the hook-lengths of the k partitions λ^{0}, …, λ^{k−1}, and by |λ^{(k)}| the sum of their weights. Set n = kq + r, 0 ≤ r < k. If r ≠ |λ_{(k)}|, then K_{λ, (1^{n})}(ζ) = 0, otherwise,

K_{λ, (1^{n})}(ζ) = (−1)^{(k−1)q}ϵ_{k}(λ/λ_{(k)}) \frac{|λ^{(k)}|!}{H(λ^{(k)})} K_{λ_{(k)}, 1^{r}}(ζ) .

Indeed, the correspondence we have just described between k-ribbon tableaux and k-uples of Young tableaux shows at once, in view of the classical hook-formula [JK], that

K_{λ/λ_{(k)}, 1^{q}}^{(k)} = \frac{|λ^{(k)}|!}{H(λ^{(k)})} .

10 Schur P -functions

As mentionned before, the specialization of Hall-Littlewood functions at q = −1 is of particular interest, since, for a strict partition λ in π_{2}(n), the symmetric functions P_{λ}(X; −1) and Q_{λ}(X; −1) coincide with those introduced by Schur in his study of the projective representations of S_{n}.

A strict partition λ of length m can be represented in the form λ = ρ_{m} + ν, where ρ_{m} = (m, m−1, …, 1). It is then known that, if X is a finite set of variables of cardinality ≤ m + 1, the following factorization holds:

P_{ρ_{m}+ν}(X; −1) = P_{ρ_{m}}(X; −1) s_{ν}(X) = s_{ρ_{m}}(X) s_{ν}(X) .  (41)

This formula due to Stanley has been used by Pragacz to show that the Z-linear combinations of Schur P -functions are characterized among symmetric polynomials by a cancellation property [Pr]. The reader is referred to [Pr] for some geometric applications.

Another interesting property of P -functions is the staircase multiplication formula [W]

P_{ρ_{r}} P_{ρ_{s}} = P_{ρ_{r}+ρ_{s}} ,  (42)

valid irrespective of the number (finite or infinite) of variables. The following identity, established in [LLT3], provides a common generalization of (41) and (42). It gives a quadratic expression of any P -function in terms of the S-functions.

Theorem 10.1 Let λ be a partition of length at most n. Then

P_{ρ_{n}+λ} = ∑_{μ} ω (s_{ρ_{n}+2μ}) D_{p_{2}∘s_{μ}}s_{λ}  (43)

the summation being over all partitions μ.

Example 10.2 With λ = (4, 3, 1) we have

P_{752} = P_{(321)+(431)} = s_{321} s_{431} + s_{32111} s_{411} + s_{3211111} (−s_{4}) + s_{32221} s_{211}
+s_{3222111} (−s_{2}) + s_{3222221} (−s_{0}) .

When λ = ρ_{m} it is well known that

\frac{∂}{∂p_{2i}} s_{λ} = 0

for all i, so that D_{p_{2}∘s_{μ}}s_{λ} = 0 for μ ≠ 0, and since s_{λ} = P_{ρ_{m}}, (43) reduces to (42). Also, when dealing with a set of variables X of cardinality n + 1, one has

ω (s_{ρ_{n}+2μ}) (X) = 0

as soon as μ ≠ 0, and (43) reduces to (41).

Other interesting special cases can be explicited. For example, when λ = (i) is a row partition,

P_{ρ_{n}+(i)} = ∑_{0≤j≤[i/2]} ω(s_{ρ_{n}+(2j)}) h_{i−2j} .  (44)

More generally, in the case where λ = ρ_{m} + 2ν with ℓ(ν) ≤ m, the derivatives can be expressed as

D_{p_{2}∘s_{μ}} s_{ρ_{m}+2ν} = ∑_{α}⟨s_{ν} , s_{μ} s_{α}⟩ s_{ρ_{m}+2α}  (45)

so that, for ℓ(ν) ≤ m ≤ n,

P_{ρ_{n}+ρ_{m}+2ν} = ∑_{μ, α} c_{μ, α}^{ν} ω(s_{ρ_{n}+2μ}) s_{ρ_{m}+2α}  (46)

where the c_{μ, α}^{ν} are the Littlewood-Richardson multiplicities.

In the general case, the multiplicity

⟨D_{p_{2}∘s_{μ}} s_{λ} , s_{γ}⟩ = ⟨s_{λ} , s_{γ} [p_{2} ∘ s_{μ}]⟩

of the product ω(s_{ρ_{n}+2μ}) s_{γ} in (43) can be combinatorially described by means of an analogue of the Littlewood-Richardson rule, in which the usual Young tableaux are to be replaced by 2-ribbons tableaux (or domino tableaux) [CL].

Surprisingly, the Schur P -functions happen to be connected to a family of orthogonal polynomials known as Bessel polynomials. These polynomials can be defined as follows. The continued fraction expansion of tanh z^{−1} has for successive convergents

F_{1} = \frac{1}{z} , F_{2} = \frac{3z}{3z^{2} + 1} , F_{3} = \frac{15z^{2} + 1}{15z^{3} + 6z} , F_{4} = \frac{105z^{3} + 10z}{105z^{4} + 45z^{2} + 1} , …

The sum of the numerator and denominator of F_{n} is a polynomial y_{n}(z) of degree n, called the nth Bessel polynomial.

A symmetric function F (X; T) (where T is some set of variables) is said to be a symmetric analogue of a formal power series f(T) if there exists a specialization X = A of X such that F (A; T) = f(T). The set of variables A can be virtual which means that the specialization is defined by specifying (in an arbitrary way) the values of a sequence of algebraically independent generators of Sym. For example, σ_{t}(X) is a symmetric analogue of the exponential function e^{t} = σ_{t}(E), where the virtual set E is defined by h_{n}(E) = 1/n!, or equivalently by p_{1}(E) = 1 and p_{k}(E) = 0 for k > 1.

A more substantial example is given by the symmetric Eulerian polynomials A_{n}(X; t), with generating series [De2]

A_{X}(t) = ∑_{n≥0} A_{n}(X; t) = \frac{1 − t}{1 − t σ_{1−t}(X)} .

The usual Eulerian polynomials are then given by

A_{n}(t) = n! A_{n}(E; t) .

The Schur P or Q-functions provide symmetric analogues of the Bessel polynomials. Set once for all q = −1, and

Y_{n}(X; z) = \frac{Q_{ρ_{n}}(X + z)}{Q_{ρ_{n}}(X)}

(in λ-ring notation). Then the vertex operator formula (23) shows that if μ is the linear functional defined by

μ(z^{n}) = (−1)^{n+1}Q_{n+1}(X) ,

then [LT],

Theorem 10.3 1. The Y_{n}(X; z) are orthogonal for μ, and more precisely,

μ(z^{k}Y_{n}(X; z)) = (−1)^{k+1} \frac{Q_{ρ_{n}∨(k+1)}(X)}{Q_{ρ_{n}}(X)} ,

which is zero for 0 ≤ k ≤ n − 1.

2. Y_{n}(E; z) = y_{n}(z)

The staircase multiplication formula (42) written in the form

Q_{ρ_{m}} Q_{ρ_{n}} = 2^{m} Q_{ρ_{m}+ρ_{n}}

(m ≤ n), gives in view of (23)

μ(z^{k} Y_{m}(X; z) Y_{n}(X; z)) = (−1)^{k+1} \frac{Q_{ρ_{m}+ρ_{n}∨(k+1)}(X)}{Q_{ρ_{m}+ρ_{n}}(X)} ,

and specializing again to X = E, one obtains a formula conjectured by Favreau in his thesis [Fa] for computing the linearization coefficients of Bessel polynomials. (This was proved independently by J. Zeng).

References

[CL] C. Carré and B. Leclerc, Splitting the square of a Schur function into its symmetric and antisymmetric parts, Institut Gaspard Monge, preprint, 1993.

[CT] C. Carré and J.-Y. Thibon, Plethysm and vertex operators, Adv. in Applied Math. 13 (1992), 390-403.

[Co] E. Cohen, A class of arithmetical functions, Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 939-944.

[DCP] C. De Concini and C. Procesi, Symmetric functions, conjugacy classes and the flag variety, Invent. Math. 64 (1981), 203-230.

[De1] J. Désarménien, Etude modulo n des statistiques mahonniennes, Actes du 22^{e} séminaire Lotharingien de Combinatoire, IRMA, Strasbourg (1990), 27-35.

[De2] J. Désarménien, Fonctions symétriques associées à des suites classiques de nombres, Ann. Sci. Éc. Norm. Sup. 16 (1983), 271-304.

[Fa] L.D. Faddeev Integrable models in 1 + 1 dimensional quantum field theory, in Les Houches Lectures 1982, Elsevier, Amsterdam, 1984.

[Fav] L. Favreau, Combinatoire des tableaux oscillants et des polynômes de Bessel, Publ. L.A.C.I.M., U.Q.A.M. Montréal, 1991.

[FS] S. Fomin and D. Stanton, Rim hook lattices, Mittag-Leffler institute, preprint No 23, 1991/92.

[Fo1] H.O. Foulkes, A survey of some combinatorial aspects of symmetric functions, in Permutations, Gauthier-Villars, Paris, 1974.

[Fo2] H.O. Foulkes, Characters of symmetric groups induced by characters of cyclic subgroups, in Combinatorics (Proc. Conf. Comb. Math. Inst. Oxford 1972), Inst. Math. Appl., Southend-on-Sea, 1972, 141-154.

[Gr] J.A. Green, The characters of the finite general linear groups, Trans. Amer. Math. Soc. 80 (1955), 402-447.

[Gu1] R.K. Gupta, Generalized exponents via Hall-Littlewood symmetric functions, Bull. Amer. Math. Soc. 16 (1987), 287-291.

[Gu2] R.K. Gupta, Characters and the q-analog of weight multiplicity, J. London Math. Soc. 36 (1987), 68-76.

[Gu3] R.K. Gupta, Problem 9, in Combinatorics and Algebra, C. Greene ed. 1984, Contemporary Mathematics vol. 34.

[Ha] P. Hall, The algebra of partitions, Proc. 4th Canadian Math. Congress, 1957.

[Han] G.-N. Han, Croissance des polynômes de Kostka, C. R. Acad. Sci. Paris, 311 (1990) 269-272.

[HW] G.M. Hardy and E.M. Wright, An introduction to the theory of numbers, (5th ed.), Oxford, Clarendon Press, 1979.

[He] W.H. Hesselink, Characters of the nullcone, Math. Ann. 252 (1980), 179-182.

[HS] R. Hotta and T.A. Springer, A specialization theorem for certain Weyl groups, Invent. Math 41 (1977), 113-127.

[JK] G. D. James and A. Kerber, The representation theory of the symmetric group, Addison-Wesley, 1981.

[Jo] K. Johnsen, On a forgotten note by Ernst Steinitz in the theory of Abelian groups, Bull. London Math. Soc. 14 (1982), 353-355.

[Ka] S. Kato, Spherical functions and a q-analogue of Kostant’s weight multiplicity formula, Invent. Math. 66 (1982), 461-468.

[Kac] V. Kac, Infinite dimensional Lie algebras, Cambridge Univ. Press, 3rd ed., 1990.

[Ke] A. Kerber, Algebraic combinatorics via finite group actions, Mannheim, 1991.

[KSW] A. Kerber, F. Sänger and B. Wagner, Quotienten und Kerne von Young-Diagrammen, Brettspiele und Plethysmen gewöhnlicher irreduzibler Darstellungen symmetrischer Gruppen, Mitt. Math. Sem. Giessen, 149 (1981), 131-175.

[KKR] S. Kerov, A.N. Kirillov and N. Yu. Reshetikhin, Combinatorics, Bethe Ansatz, and representations of the symmetric group, J. Sov. Math., 41 (1988), 916-924.

[Ki1] A.N. Kirillov, Combinatorial identities and completeness of states of Heisenberg magnetics, in Questions of quantum field theory and statistical physics 4, J. Sov. Math. 30 No. 4 (1985).

[Ki2] A.N. Kirillov, Completeness of states of generalized Heisenberg magnetics, in Automorphic functions and number theory 11, J. Sov. Math. 36 No. 1 (1987).

[Ki3] A. N. Kirillov, On the Kostka-Green-Foulkes polynomials and Clebsch-Gordan numbers, J. of Geometry and Physics, 5:3 (1988), 365-389.

[Ki4] A.N. Kirillov, Decomposition of symmetric and exterior powers of the adjoint representation of 𝔤𝔩_{N}, Advanced Series in Math. Phys., 16 B (1992), 545-580.

[Ki5] A.N. Kirillov, Unimodality of generalized Gaussian coefficients, C. R. Acad. Sci. Paris, 315 (1992), 497-501.

[KR] A. N. Kirillov and N. Yu. Reshetikhin, Bethe ansatz and the combinatorics of Young tableaux, J. Sov. Math., 41 (1988), 925-955.

[Kn] D.E. Knuth, Permutations, matrices and generalized Young tableaux, Pacific. J. Math. 34 (1970), 709-727.

[Ko] B. Kostant, Lie group representations on polynomial rings, Amer. J. Math. 85 (1963), 327-404.

[KW] W. Kraskiewicz and J. Weyman, Algebra of invariants and the action of a Coxeter element, Preprint Math. Inst. Univ. Copernic, Toruń, Poland.

[La] A. Lascoux, Cyclic permutations on words, tableaux and harmonic polynomials, Proc. of the Hyderabad conference on algebraic groups, 1989, Manoj Prakashan, Madras (1991), 323-347.

[LLT1] A. Lascoux, B. Leclerc and J.Y. Thibon, Fonctions de Hall-Littlewood et polynômes de Kostka-Foulkes aux racines de l’unité, C.R. Acad. Sci. Paris, 316 (1993), 1-6.

[LLT2] A. Lascoux, B. Leclerc and J.Y. Thibon, Green polynomials and Hall-Littlewood functions at roots of unity, Europ. J. Combinatorics 15 (1994), 173-180.

[LLT3] A. Lascoux, B. Leclerc and J.Y. Thibon, Une nouvelle expression des fonctions P de Schur, C.R. Acad. Sci. Paris, 316 (1993), 221-224.

[LS1] A. Lascoux and M. P. Schützenberger, Le monoïde plaxique, in “Noncommutative structures in algebra and geometric combinatorics” (A. de Luca Ed.), Quaderni della Ricerca Scientifica del C. N. R., Roma, 1981.

[LS2] A. Lascoux and M. P. Schützenberger, Sur une conjecture de H.O. Foulkes, C.R. Acad. Sci. Paris 286A (1978), 323-324.

[LT] B. Leclerc and J.- Y. Thibon, Analogues symétriques des polynômes de Bessel, C.R. Acad. Sci. Paris, 315 (1992), 527-530.

[Li1] D. E. Littlewood, On certain symmetric functions, Proc. London Math. Soc. 43 (1961), 485-498.

[Li2] D. E. Littlewood, The theory of group characters and matrix representations of groups, Oxford, 1950 (second edition).

[Li3] D. E. Littlewood, Modular representations of symmetric groups, Proc. Roy. Soc. A. 209 (1951) 333-353

[Lu1] G. Lusztig, Singularities, character formulas, and a q-analog of weight multiplicities, Analyse et topologie sur les espaces singuliers (II-III), Astérisque 101-102 (1983), 208-227.

[Lu2] G. Lusztig, Green polynomials and singularities of unipotent classes, Adv. in Math. 42 (1981), 169-178.

[Mcd] I. G. Macdonald, Symmetric functions and Hall polynomials, Oxford, 1979.

[MT] A.I. Molev and L.M. Tsalenko, Representations of symmetric groups in a free Lie (super) algebra and in the space of harmonic polynomials, Funct. Anal. Appl. 20:2 (1986), 150-152.

[Mo1] A.O. Morris, The characters of the group GL(n, q), Math. Zeitschr. 81 (1963), 112-123.

[Mo2] A.O. Morris, On an algebra of symmetric functions, Quart. J. Math. Oxford Ser. (2) 16 (1965), 53-64.

[Mo3] A.O. Morris, The multiplication of Hall functions, Proc. Lond. Math. Soc. 13 (1963), 733-742.

[MS] A. O. Morris and N. Sultana, Hall-Littlewood polynomials at roots of 1 and modular representations of the symmetric group, Math. Proc. Cambridge Phil. Soc., 110 (1991), 443-453.

[Mu] T. Muir, A treatise on the theory of determinants, Macmillan, London, 1882.

[NV] C.A. Nicol and H.S. Vandiver, A Von Sterneck arithmetical function and restricted partitions with respect to a modulus, Proc. Nat. Acad. Sci. U.S.A. 40 (1954), 825-835.

[Pr] P. Pragacz, Algebro-geometric applications of Schur S- and P - polynomials in Topics in invariant theory (Séminaire Dubreil-Malliavin), Lecture Notes in Math. vol. 1478, Springer, 1991.

[ST] T. Scharf and J.-Y. Thibon, A Hopf algebra approach to inner plethysm, Adv. in Math. (to appear).

[S] I. Schur, Über die Darstellung der symmetrischen und der alternierenden Gruppen durch gebrochene lineare Substitutionen, J. reine angew. Math. 139 (1911), 155-250.

[Sc] M.P. Schützenberger, Propriétés nouvelles des tableaux de Young, Séminaire Delange-Pisot-Poitou, 19ème année, 26, 1977/78.

[SW] D. Stanton and D. White, A Schensted algorithm for rim-hook tableaux, J. Comb. Theory A 40, (1985), 211-247.

[Sta] R.P. Stanley, The stable behaviour of some characters of SL(n, C), Linear and Multilinear Alg. 16 (1984), 3-27.

[St] E. Steinitz, Zur theorie der Abel’schen gruppen, Jahresberichta der DMV 9 (1901), 80-85.

[Su] N. Sultana, Hall-Littlewood functions and their applications to representation theory, Thesis, University of Wales, 1990.

[TF] L.A. Takhtadzhyan and L.D. Faddeev, Spectrum and scattering of stimuli of excitations in the one-dimensional isotropic Heisenberg model, Zap. Nauch. Semin. LOMI, 109 (1981), 134-173.

[T] J.-Y. Thibon, Hopf algebras of symmetric functions and tensor products of symmetric group representations, Internat. J. of Alg. Comp., 1 (1991), 207-221.

[W] D.R. Worley, A theory of shifted Young tableaux, Thesis, M.I.T. 1984.