RIBBON TABLEAUX, HALL-LITTLEWOOD FUNCTIONS AND UNIPOTENT VARIETIES ^{∗}

Alain Lascoux^{†}, Bernard Leclerc^{†} and Jean-Yves Thibon^{‡}

Abstract

We introduce a new family of symmetric functions, which are defined in terms of ribbon tableaux and generalize Hall-Littlewood functions. We present a series of conjectures, and prove them in two special cases.

1 Introduction

Hall-Littlewood functions [Li1] are known to be related to a variety of topics in representation theory, geometry and combinatorics. These symmetric functions arise in the character theory of finite linear groups [Gr], in the geometry of unipotent varieties [Sh1, HSh], in particular as characteristics of the representations of the symmetric group in their cohomology [HS], and appear to be related to the Quantum Inverse Scattering Method [KR]. From a combinatorial point of view, their description involves the deepest aspects of the theory of Young tableaux: the multiplicative structure (plactic monoid) and the ordered structure derived from the cyclage operation [LS2, La]. There exists also a description in terms of Kashiwara’s theory of crystal bases [LLT4].

Another kind of application of Hall-Littlewood functions is concerned with the representation theory of the complex linear group GL(n, C). The general setting is the following. Suppose we are given a finite dimensional representation V of G = GL(n, C). The symmetric group S_{k} acts (on the right) on the tensor space W = V^{⊗k} by

v_{1} ⊗ v_{2} ⊗ ⋯ ⊗ v_{k} · σ = v_{σ(1)} ⊗ v_{σ(2)} ⊗ ⋯ ⊗ v_{σ(k)} .

Let γ ∈ S_{k} be a k-cycle. The eigenvalues of γ, as an endomorphism of W are ζ^{r}, r = 0, …, k − 1, where ζ is a primitive k-th root of unity.

∗Partially supported by PRC Math-Info and EEC grant n^{0} ERBCHRXCT930400

†L.I.T.P., Université Paris 7, 2 place Jussieu, 75251 Paris cedex 05, France

‡Institut Gaspard Monge, Université de Marne-la-Vallée, 2 rue de la Butte-Verte, 93166 Noisyle-Grand cedex, France

Let W^{(i)} ⊂ V^{⊗k} be the eigenspace of γ corresponding to the eigenvalue ζ^{i}. Then, W^{(i)} is a sub-G-module of W, and an important problem is to compute its decomposition into irreducibles from the character of V. In terms of symmetric functions, this is a plethysm problem: if F = ch V is the formal character of V and

ℓ_{k}^{(i)} = ch [(C^{n})^{⊗k}]^{(i)} = \frac{1}{k} ∑_{d|k} c(i, d)p_{d}^{k/d}

is the character of W^{(i)} in the special case where V = C^{n} (the basic representation of G), the problem is to expand the plethysm ℓ_{k}^{(i)} ∘ F as a linear combination of Schur functions

ℓ_{k}^{(i)} ∘ F = ∑_{λ} c_{λ}s_{λ} .

The coefficients c_{λ} are nonnegative integers, for which it is rather unlikely that a closed formula could exist, and a combinatorial interpretation of these coefficients, allowing their individual computation, should be considered as a satisfactory solution of the problem.

This problem is solved in [LLT1, LLT2], in the case where V is a tensor product of exterior or symmetric powers of the basic representation, i.e. V = Λ^{μ_{1}}C^{n} ⊗ ⋯ ⊗ Λ^{μ_{r}}C^{n}, or V = S^{μ_{1}}C^{n} ⊗⋯⊗S^{μ_{r}}C^{n}. The answer in this case is that the character of W^{(i)} can be obtained by reducing modulo 1 − q^{k} a certain Hall-Littlewood function. The required combinatorial interpretation is then obtained from the one of Hall-Littlewood functions.

Another case which is completely solved is when k = 2 and V = V_{λ} is an irreducible representation [CL]. This involves a new version of the Littlewood-Richardson rule, where ordinary tableaux are replaced by domino tableaux. This rule can also be formulated as a property of the reduction modulo 1 − q^{2} of certain symmetric functions, also defined in terms of domino tableaux, and depending on a parameter q [CL, KLLT].

In this paper, we introduce a new family of symmetric functions H_{λ}^{(k)}(x; q), defined as generating functions of certain sets of k-ribbon (or rim-hook) tableaux according to a statistic called spin, which contains the Hall-Littlewood functions and the domino functions as particular cases, and continue to display the same type of behaviour, at least at the experimental level.

We present a series of conjectures on these new functions, and prove them in the two extreme cases: for the shortest possible ribbons (dominoes), and for sufficiently long ones (the stable case).

The methods used for proving these two cases are quite different. In the domino case, the proofs are entierely combinatorial, and do not seem to be generalizable to other cases. This is probably due to the fact that the hyperoctahedral group is the only group in the series S_{n}[C_{k}], k ≥ 2 (wreath products of a cyclic group by a symmetric group) which is also a Weyl group. In the stable case, the proofs rely upon the cell decompositions of unipotent varieties [Sh1, HSh]. We believe that the intermediate cases may also be proved by similar techniques, in terms of the geometry of other subvarieties of flag manifolds.

2 Hall-Littlewood functions and unipotent varieties

Our notations for symmetric functions will be essentially those of the book [Mcd], to which the reader is referred for more details.

Let X = {x_{1}, x_{2}, …} be an infinite set of indeterminates and Sym be the ring of symmetric functions in X, with coefficients in C(q), q being another indeterminate. In what follows, the scalar product ⟨ , ⟩ on Sym will always be the standard one, for which Schur functions form an orthonormal basis. We denote by Q′_{μ}(X; q) the image of the Hall-Littlewood function Q_{μ}(X; q) by the ring homomorphism p_{k} ↦ (1 − q^{k})^{−1}p_{k}. That is, (Q′_{μ}) is the adjoint basis of the basis (P_{μ}) for the standard scalar product, and in λ-ring notation, Q′_{μ}(X; q) = Q(X/(1 − q); q). In the Schur basis,

Q′_{μ}(X; q) = ∑_{λ} K_{λμ}(q)s_{λ}(X)  (1)

where the K_{λμ}(q) are the Kostka-Foulkes polynomials. The polynomial K_{λμ}(q) is the generating function of a statistic c called charge on the set Tab (λ, μ) of Young tableaux of shape λ and weight μ

K_{λμ}(q) = ∑_{t∈Tab (λ,μ)} q^{c(t)} .  (2)

We shall also need the Q̃′-functions, defined by

Q̃′_{μ}(X; q) = ∑_{λ} K̃_{λμ}(q)s_{λ}(X) = q^{n(μ)}Q′_{μ}(X; q^{−1}) .  (3)

The polynomial K̃_{λμ}(q) is the generating function of the complementary statistic c̃(t) = n(μ) − c(t), which is called cocharge. The operation of cyclage endows Tab (λ, μ) with the structure of a rank poset, in which the rank of a tableau is equal to its cocharge (see [La]).

When the parameter q is interpreted as the cardinality of a finite field F_{q}, it is known that K̃_{λμ}(q) is equal to the value χ^{λ}(u) of the unipotent character χ^{λ} of G = GL_{n}(F_{q}) on a unipotent element u with Jordan canonical form specified by the partition μ (see [Lu2]).

In this specialization, the coefficients

G̃_{νμ}(q) = ⟨h_{ν} , Q̃′_{μ}⟩  (4)

of the Q̃′-functions on the basis of monomial symmetric functions are also the values of certain characters of G on unipotent classes. Let P_{ν} denote a parabolic subgroup of type ν of G, for example the group of upper block triangular matrices with diagonal blocks of sizes ν_{1}, …, ν_{r}, and consider the permutation representation of G over C[G/P_{ν}]. The value ξ^{ν}(g) of the character ξ^{ν} of this representation on an element g ∈ G is equal to the number of fixed points of g on G/P_{ν}. Then, it can be shown that, for a unipotent u of type μ,

ξ^{ν}(u) = G̃_{νμ}(q) .  (5)

The factor set G/P_{ν} can be identified with the variety F_{ν} of ν-flags in V = F_{q}^{n}

V_{ν_{1}} ⊂ V_{ν_{1}+ν_{2}} ⊂ … ⊂ V_{ν_{1}+...ν_{r}} = V

where dim V_{i} = i. Thus, G̃_{νμ}(q) is equal to the number of F_{q}-rational points of the algebraic variety F_{ν}^{u} of fixed points of u in F_{ν}.

It has been shown by N. Shimomura ([Sh1], see also [HSh]) that the corresponding complex variety F_{ν}^{u}[C] admits a cell decomposition, involving only cells of even real dimensions. More precisely, this cell decomposition is a partition in locally closed subvarieties, each being algebraically isomorphic to an affine space. Thus, the odd-dimensional homology groups are zero, and if

Π_{νμ}(t^{2}) = ∑_{i} t^{2i}dim H_{2i}(F_{ν}^{u}, Z)

is the Poincaré polynomial of F_{ν}^{u}, one has |F_{ν}^{u}| = Π_{νμ}(q). But this is also equal to G̃_{νμ}(q), and as this is true for an infinite set of values of q, one has Π_{νμ}(z) = G̃_{νμ}(z) as polynomials. That is, the coefficient of Q̃′_{μ} on the monomial function m_{ν} is the Poincaré polynomial of F_{ν}^{u}, for a unipotent u of type μ.

Writing

Q̃′_{μ} = ∑_{λ,ν} K̃_{λμ(q)}K_{λν} m_{ν} ,  (6)

one sees that

G̃_{νμ}(q) = ∑_{(t_{1},t_{2})∈Tab (λ,μ)×Tab (ν,μ)} q^{c̃(t_{1})} .  (7)

Knuth’s extension of the Robinson-Schensted correspondence [Kn] is a bijection between the set

∐_{λ} Tab (λ, μ) × Tab (λ, ν)

of pairs of tableaux with the same shape, and the double coset space S_{μ}\S_{n}/S_{ν} of the symmetric group S_{n} modulo two parabolic subgroups. Double cosets can be encoded by two-line arrays, integer matrices with prescribed row and column sums, or by tabloids.

Let ν and μ be arbitrary compositions of the same integer n. A μ-tabloid of shape ν is a filling of the diagram of boxes with row lengths ν_{1}, ν_{2}, …, ν_{r}, the lowest row being numbered 1 (French convention for tableaux), such that the number i occurs μ_{i} times, and such that each row is nondecreasing. For example,

3 1 1 1 1 1 3 2 3

is a (5, 1, 3)-tabloid of shape (2, 3, 3, 1).

We denote by L(ν, μ) the set of tabloids of shape ν and weight μ. A tabloid will be identified with the word obtained by reading it from left to right and top to bottom. Then,

G̃_{νμ}(q) = ∑_{T ∈L(ν,μ)} q^{c̃(T)} .  (8)

Example 2.1 To compute G̃_{42,321}(q) one lists the elements of L((4, 2), (3, 2, 1)), which are

2 3 1 1 1 2  2 2 1 1 1 3  1 3 1 1 2 2  1 2 1 1 2 3  1 1 1 2 2 3

Reading them as prescribed, we obtain the words

231112 221113 131122 121123 111223

whose respective charges are 2, 1, 3, 2, 4. The cocharge polynomial is thus G̃_{42,321}(q) = 1 + q + 2q^{2} + q^{3}.

In Shimomura’s decomposition of the fixed point variety F_{μ}^{u} of a unipotent of type ν, the cells are indexed by tabloids of shape ν and weight μ. The dimension d(T) of the cell c_{T} indexed by T ∈ L(ν, μ) is computed by an algorithm described below, and gives another combinatorial interpretation of the polynomial G̃_{μν}(q), exchanging the rôles of shape and weight:

G̃_{μν} = ∑_{T ∈L(μ,ν)} q^{c̃(T)} = ∑_{T ∈L(ν,μ)} q^{d(T)} .  (9)

The dimensions d(T) are given by the following algorithm.

1. If T ∈ L(ν, (n)) then d(T) = 0;

2. If μ = (μ_{1}, μ_{2}) has exactly two parts, and T ∈ L(ν, μ), then d(T) is computed as follows. A box α of T is said to be special if it contains the rightmost 1 of its row. For a box β of T, put d(β)=0 if β does not contain a 2, and if β contains a 2, set d(β) equal to the number of nonspecial 1’s lying in the column of β, plus the number of special 1’s lying in the same column, but in a lower position. Then

d(T) = ∑_{β} d(β) .

3. Let μ = (μ_{1}, …, μ_{k}) and μ^{∗} = (μ_{1}, …, μ_{k−1}). For T ∈ L(ν, μ), let T_{1} be the tabloid obtained by changing the entries k into 2 and all the other ones into 1. Let T_{2} be the tabloid obtained by erasing all the entries k, and rearranging the rows in the appropriate order. Then,

d(T) = d(T_{1}) + d(T_{2}) .  (10)

Example 2.2 With T = 1 4 1 2 3 1 1 2 ∈ L(332, 4211), one has

T_{1} := 1 2 1 1 1 1 1 1  T_{2} = 1 1 2 3 1 1 2  T_{21} = 1 1 1 2 1 1 1  T_{22} = 1 1 2 1 1 2

where the special entries are printed in boldtype. Thus, d(T) = t(T_{1}) + d(T_{2}) = 2 + d(T_{21}) + d(T_{22}) = 4.

We shall need a variant of this construction, in which the shape ν is allowed to be an arbitrary composition, and where in step 3, the rearranging of the rows is supressed. Such a variant has already been used by Terada [Te] in the case of complete flags.

That is, we associate to a tabloid T ∈ L(ν, μ) an integer e(T), defined by

1. For T ∈ L(ν, (n)), e(T) = d(T) = 0;

2. For T ∈ L(ν, (μ_{1}, μ_{2})), e(T) = d(T);

3. Otherwise e(T) = e(T_{1}) + e(T_{2}) where T_{1} is defined as above, but this time T_{2} is obtained from T by erasing the entries k, without reordering.

Lemma 2.3 Let λ = (λ_{1}, …, λ_{r}) be a partition, and let ν = λ·σ = (λ_{σ(1)}, …, λ_{σ(r)}), σ ∈ S_{r}. Then, the distribution of e on L(ν, μ) is the same as the distribution of d on L(λ, μ). That is,

D_{λμ}(q) = ∑_{T ∈L(λ,μ)} q^{d(T)} = E_{νμ}(q) = ∑_{T ∈L(ν,μ)} q^{e(T)} .

In particular, D_{λμ}(q) = E_{λμ}(q).

Proof — This could be proved by repeating word for word the geometric argument of [Sh1]. We give here a short combinatorial argument. As the two statistics coincide on tabloids whose shape is a partition and whose weight has at most two parts, the only thing to prove, thanks to the recurrence formula, is that e has the same distribution on L(β, (μ_{1}, μ_{2})) as on L(α, (μ_{1}, μ_{2})) when β is a permutation of α. The symmetric group being generated by the elementary transpositions σ_{i} = (i, i+1), one may assume that β = ασ_{i}. We define the image T σ_{i} of a tabloid T ∈ L(α, (μ_{1}, μ_{2})) by distinguishing among the following configurations for rows i and i + 1:

1.

\begin{matrix}x_{1} & … & x_{k} & 2 & 2^{r} \\ 1 & … & 1 & 1 & 2^{s} \end{matrix}  \underrightarrow{σ_{i}}  \begin{matrix}x_{1} & … & x_{k} & 2 & 2^{s} \\ 1 & … & 1 & 1 & 2^{r} \end{matrix}

2.

\begin{matrix}1 & … & 1 & 1 & 2^{r} \\ x_{1} & … & x_{k} & 2 & 2^{s} \end{matrix}  \underrightarrow{σ_{i}}  \begin{matrix}1 & … & 1 & 1 & 2^{s} \\ x_{1} & … & x_{k} & 2 & 2^{r} \end{matrix}

3. In all other cases, the two rows are exchanged:

\begin{matrix}x_{1} & … & x_{r} \\ y_{1} & … & & y_{s} \end{matrix}  \underrightarrow{σ_{i}}  \begin{matrix}y_{1} & … & & y_{s} \\ x_{1} & … & x_{r} \end{matrix}

From this definition, it is clear that e(T σ_{i}) = e(T). Moreover, it is not difficult to check that this defines an e-preserving action of the symmetric group S_{m} on the set of μ-tabloids with m rows, such that L(α, μ)σ = L(ασ, μ) (the only point needing a verification is the braid relation σ_{i}σ_{i+1}σ_{i} = σ_{i+1}σ_{i}σ_{i+1}.

Thus, for a partition λ and a two-part weight μ = (μ_{1}, μ_{2}), d and e coincide on L(λ, μ), and for σ ∈ S_{m}, E_{λσ,μ}(q) = D_{λμ}(q). Now, by induction, for μ = (μ_{1}, …, μ_{k}),

D_{λμ}(q) = ∑_{T ∈L(λ,μ)} q^{d(T_{1})}q^{d(T_{2})}

= ∑_{λ̄=shape (T_{1})} q^{d(T_{1})}D_{λ̄,μ^{∗}}(q) = ∑_{λ̄=shape (T_{1})} e^{e(T_{1})}E_{λ̄,μ^{∗}}(q) = E_{λμ}(q) .
□

Example 2.4 Take λ = (3, 2, 1), μ = (4, 2) and ν = λσ_{1}σ_{2} = (3, 1, 2). The μtabloids of shape λ are

\begin{matrix}T & 2 1 2 1 1 1 & 2 1 1 1 1 2 & 1 1 1 1 2 2 & 1 2 2 1 1 1 & 1 1 2 1 1 2 \\ d(T) & 3 & 2 & 0 & 2 & 1 \end{matrix}

The ν-tabloids of shape λ are

\begin{matrix}T & 1 1 1 1 2 2 & 1 1 1 2 1 2 & 1 1 2 1 1 2 & 1 1 2 2 1 1 & 1 2 2 1 1 1 \\ e(T) & 2 & 3 & 0 & 2 & 1 \end{matrix}

Thus, D_{λμ}(q) = E_{νμ}(q) = 1 + q + 2q^{2} + q^{3} = G̃_{μλ}(q). The tabloids contributing a term q^{2} are apparied in the following way:

2 1 1 1 1 2 ⟶ 1 1 2 2 1 1  1 2 2 1 1 1 ⟶ 1 1 1 1 2 2

Remark The only property that we shall need in the sequel is the equality D_{λμ}(q) = E_{λμ}(q). However, it is possible to be more explicit by constructing a bijection exchanging d and e. The above action of S_{m} can be extended to tabloids with arbitrary weight, still preserving e. Suppose for example that we want to apply σ_{i} to a tabloid T whose restriction to rows i, i + 1 is

1 1 2 3 7 7 9 1 1 1 2 6 6 6 8 8 9

One first determines the positions of the greatest entries, which are the 9’s, in T σ_{i}. Starting with an empty diagram of the permuted shape (10, 7), one constructs T_{1} as above by converting all the entries 9 of T into 2 and the remaining ones into 1. Then we apply σ_{i} to T_{1}, and the positions of the 2 in T_{1}σ_{i} give the positions of the 9 in T σ_{i}. Then, the entries 9 are removed from T ad the procedure is iterated until one reaches a tabloid whose rows i and i + 1 are of equal lenghts. This tabloid is then copied (without permutation) in the remaining part of the result. On the example, this gives

1 1 1 1 1 1 2 1  9 1 1 1 1 1 1 1 1 2  1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 2  9 9 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1  1 1 1 1 1 1 1 2 2 1 1 1 1 1 1  9 9 8 8 1 1 1 1 2 2 1 1 1 1 1 1 1  1 1 1 1 1 1 1 1 1 1 1 2 2  7 7 9 8 8 9 1 1 1 1 1 1 1 1 2 2 2  1 1 1 1 2 2 2 1 1 1 1  6 6 6 8 8 9 7 7 9 1 1 1 2 1 1 1 1  1 1 1 2 1 1 1 1  1 1 2 3 6 6 6 8 8 9 1 1 1 2 7 7

3 Specializations at roots of unity

As recalled in the preceding section, the Hall-Littlewood functions with parameter specialized to the cardinality q of a finite field F_{q} provide information about the characters of the linear group GL(n, F_{q}) over this field. It turns out that when the parameter is specialized to a complex root of unity, one obtains information about representations of GL(n, C), that is, a combinatorial decomposition of certain plethysms [LLT1, LLT2]. We give now a brief review of the main results of these papers.

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 3.1 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}} .  (11)

The functions Q′_{(i^{k})}(X; ζ) appearing in the right-hand side of (11) can be expressed as plethysms.

Theorem 3.2 Let p_{k} ∘ h_{n} denote the plethysm of the complete function h_{n} by the power-sum 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 3.3 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} .

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}).

For k, n ∈ N, the Ramanujan or Von Sternecksum 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. [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 property [Co] can be regarded as a generalization of the Moebius inversion formula:

Lemma 3.4 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} .  (12)

These functions were first encountered by Foulkes as Frobenius characteristics of the representations of the symmetric group induced by irreducible representations of transitive cyclic subgroup [Fo]. 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 [De]. Taking into account Theorems 3.1 and 3.2, one obtains:

Theorem 3.5 [LLT2] 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.

This gives as well the plethysms with product of complete functions, since

⟨s_{μ′} , ℓ_{k}^{(r)} ∘ e_{λ}⟩ = \{ \begin{matrix}⟨s_{μ} , ℓ_{k}^{(r)} ∘ h_{λ}⟩ & if |λ| is even \\ ⟨s_{μ} , ℓ̃_{k}^{(r)} ∘ h_{λ}⟩ & if |λ| is odd \end{matrix}

where ℓ̃_{k}^{(r)} = ω(ℓ_{k}^{(r)}) = ℓ_{k}^{(s)} with s = k(k − 1)/2 − r.

Example 3.6 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:

3 2 4 1 1 2 3 4  4 2 3 1 1 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 more combinatorial formulation of Theorems 3.1 and 3.2 can be presented by means of the notion of ribbon tableau, which will also provide the key for their generalization.

4 Ribbon tableaux

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 1 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

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

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 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 3.1, 3.2 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 kuples 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 1, 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}}  (13)

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 (13).

More generally, if λ is replaced by a skew partition λ/ν, (13) 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 3.1 and 3.2 in the following way:

Theorem 4.1 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) .  (14)

Example 4.2 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 (14) 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)}|, (14) 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 4.3 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 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)})} .

5 H-functions

Let λ be a partition without k-core, and with k-quotient (λ^{0}, …, λ^{k−1}). For a ribbon tableau T of weight μ, let x^{T} = x_{1}^{μ_{1}}x_{2}^{μ_{2}} ⋯ x_{r}^{μ_{r}}. Then, the Stanton-White correspondence shows that the generating function

G_{λ}^{(k)} = ∑_{T ∈Tab_{k}(λ, ·)} x^{T} = ∏_{i=0}^{k−1} ∑_{t_{i}∈Tab (λ^{i}, ·)} x^{t_{i}} = ∏_{i=0}^{k−1} s_{λ^{i}}  (15)

is a product of Schur functions. Introducing in this equation an appropriate statistic on ribbon tableaux, one can therefore obtain q-analogues of products of Schur functions. The statistic called cospin, described below, leads to q-analogues with interesting properties.

Let R be a k-ribbon, h(R) its heigth and w(R) its width.

h(R)  w(R)

The spin of R, denoted by s(R), is defined as

s(R) = \frac{h(R) − 1}{2}  (16)

and the spin of a ribbon tableau T is by definition the sum of the spins of its ribbons. For example, the ribbon tableau

12 1 2 3 5 1 2 4 2 5 4 5 6

has a spin equal to 6.

For a partition λ without k-core, let

s_{k}^{∗}(λ) = max{s(T) | T ∈ Tab_{k}(λ, ·)} .  (17)

The cospin s̃(T) of a k-ribbon tableau T of shape λ is then

s̃(T) = s_{k}^{∗}(λ) − s(T) .  (18)

Although s(T) can be a half-integer, it is easily seen that s̃(T) is always an integer. Also, there is one important case where s(T) is an integer. This is when the shape λ of T is of the form kμ = (kμ_{1}, kμ_{2}, …, kμ_{r}). In this case, the partitions constituting the k-quotient of λ are formed by parts of μ, grouped according to the class modulo k of their indices. More precisely, λ^{i} = {μ_{r} | r ≡ −i mod k}

We can now define three families of polynomials

G_{λ}^{(k)}(X; q) = ∑_{T ∈Tab_{k}(λ, ·)} q^{s̃(T)}x^{T}  (19)
H̃_{μ}^{(k)}(X; q) = ∑_{T ∈Tab_{k}(kμ, ·)} q^{s̃(T)}x^{T} = G_{kμ}^{(k)}(X; q)  (20)
H_{μ}^{(k)}(X; q) = ∑_{T ∈Tab_{k}(kμ, ·)} q^{s(T)}x^{T} = q^{s_{k}^{∗}(kμ)}H̃_{μ}^{(k)}(X; 1/q) .  (21)

The parameter k will be called the level of the corresponding symmetric functions. There is strong experimental evidence for the following conjectures.

Conjecture 5.1 (symmetry) The polynomials G̃_{λ}^{(k)}, H̃_{μ}^{(k)} and H_{μ}^{(k)} are symmetric.

Conjecture 5.2 (positivity) Their coefficients on the basis of Schur functions are polynomials with nonnegative integer coefficients.

Conjecture 5.3 (monotonicity) H_{μ}^{(k+1)} − H_{μ}^{(k)} is positive on the Schur basis.

Conjecture 5.4 (plethysm) When μ = ν^{k}, for ζ a primitive k-th root of unity,

H_{ν^{k}}^{(k)}(ζ) = (−1)^{(k−1)|ν|} p_{k} ∘ s_{ν}

Equivalently,

H_{ν^{k}}^{(k)}(q) mod 1 − q^{k} = ∑_{i=0}^{k−1} q^{k}ℓ_{k}^{(i)} ∘ s_{ν}

The following statements will be proved in the forthcoming sections.

Theorem 5.5 The difference Q′_{μ} − H_{μ}^{(2)} is nonnegative on the Schur basis.

Theorem 5.6 For k ≥ ℓ(μ), H_{μ}^{(k)} is equal to the Hall-Littlewood function Q′_{μ}.

Taking into account the results of [LLT1, LLT2] and [CL], this is sufficient to establish all the conjectures for k = 2 and k ≥ ℓ(μ).

Example 5.7 (i) The 3-quotient of λ = (3, 3, 3, 2, 1) is ((1), (1, 1), (1)) and

G̃_{33321}(q) = m_{31} + (1 + q)m_{22} + (2 + 2q + q^{2})m_{211}
+(3 + 5q + 3q^{2} + q^{3})m_{1111}
= s_{31} + qs_{22} + (q + q^{2})s_{211} + q^{3}s_{1111}

is a q-analogue of the product

s_{1}s_{11}s_{1} = s_{31} + s_{22} + 2s_{211} + s_{1111} .

(ii) The H-functions associated to the partition λ = (3, 2, 1, 1) are

H_{3211}^{(2)} = s_{3211} + q s_{322} + q s_{331} + q s_{4111}
+(q + q^{2}) s_{421} + q^{2} s_{43} + q^{2} s_{511} + q^{3} s_{52}
H_{3211}^{(3)} = s_{3211} + q s_{322} + (q + q^{2}) s_{331} + q s_{4111}
+(q + 2q^{2}) s_{421} + (q^{2} + q^{3}) s_{43} + (q^{2} + q^{3}) s_{511}
+2q^{3} s_{52} + q^{4} s_{61}
H_{3211}^{(4)} = s_{3211} + q s_{322} + (q + q^{2}) s_{331} + q s_{4111}
+(q + 2q^{2} + q^{3}) s_{421} + (q^{2} + q^{3} + q^{4}) s_{43} + (q^{2} + q^{3} + q^{4}) s_{511}
+(2q^{3} + q^{4} + q^{5}) s_{52} + (q^{4} + q^{5} + q^{6}) s_{61} + q^{7} s_{7}
= Q′_{3211}

and we see that s_{3211} < H_{3211}^{(2)} < H_{3211}^{(3)} < H_{3211}^{(4)} = Q′_{3211} .

(iii) The plethysms of s_{21} with the cyclic characters ℓ_{3}^{(i)} are given by the reduction modulo 1 − q^{3} of

H_{222111}^{(3)} = q^{9}s_{63} + (q + 1)q^{7}s_{621} + q^{6}s_{6111} + (q + 1)q^{7}s_{54} + (q^{3} + 2q^{2} + 2q + 1)q^{5}s_{531}
+(q^{2} + 2q + 1)q^{5}s_{522} + (q^{3} + 2q^{2} + 2q + 1)q^{4}s_{5211} + (q + 1)q^{4}s_{51111}
+(q^{2} + 2q + 1)q^{5}s_{441} + (q^{3} + 2q^{2} + 3q + 2)q^{4}s_{432} + (2q^{3} + 3q^{2} + 3q + 1)q^{3}s_{4311}
+(q^{3} + 3q^{2} + 3q + 2)q^{3}s_{4221} + (q^{3} + 2q^{2} + 2q + 1)q^{2}s_{42111} + q^{3}s_{411111} + (q^{3} + 1)q^{3}s_{333}
+(2q^{3} + 3q^{2} + 2q + 1)q^{2}s_{3321} + (q^{2} + 2q + 1)q^{2}s_{33111} + (q^{2} + 2q + 1)q^{2}s_{3222}
+(q^{3} + 2q^{2} + 2q + 1)qs_{32211} + (q + 1)qs_{321111} + (q + 1)qs_{22221} + s_{222111}

Indeed,

H_{222111}^{(3)} mod 1 − q^{3} = (2s_{5211} + s_{22221} + s_{321111} + 3s_{4311}
+2s_{32211} + s_{522} + 3s_{432} + 3s_{3321} + s_{33111} + s_{3222} + s_{51111}
+3s_{4221} + 2s_{531} + 2s_{42111} + s_{54} + s_{621} + s_{441})q^{2}
+(2s_{5211} + s_{22221} + s_{321111} + 3s_{4311} + 2s_{32211} + s_{522}

+3s_{432} + 3s_{3321} + s_{33111} + s_{3222} + s_{51111}
+3s_{4221} + 2s_{531} + 2s_{42111} + s_{54} + s_{621} + s_{441})q
+2s_{33111} + s_{63} + s_{6111} + 2s_{531} + 2s_{522} + 2s_{5211} + 2s_{441}
+2s_{432} + 3s_{4311} + 3s_{4221} + 2s_{42111} + s_{411111} + 2s_{333}
+2s_{3321} + 2s_{3222} + s_{222111} + 2s_{32211}
= q^{2}ℓ_{3}^{(2)} ∘ s_{21} + qℓ_{3}^{(1)} ∘ s_{21} + ℓ_{3}^{(0)} ∘ s_{21} .

6 The case of dominoes

For k = 2, the conjectures can be established by means of the combinatorial constructions of [CL] and [KLLT]. In this case, conjectures 5.1, 5.2 and 5.4 follow directly from the results of [CL], and the only point remaining to be proved is Theorem 5.5.

The important special feature of domino tableaux is that there exits a natural notion of Yamanouchi domino tableau. These tableaux correspond to highest weight vectors in tensor products of two irreducible GL_{n}-modules, in the same way as ordinary Yamanouchi tableaux are the natural labels for highest weight vectors of irreducible representations.

The column reading of a domino tableau T is the word obtained by reading the successive columns of T from top to bottom and left to right. Horizontal dominoes, which belong to two succesive columns i and i + 1 are read only once, when reading column i. For example, the column reading of the domino tableau

1 2 34 1 2

is col (T) = 431212.

A Yamanouchi word is a word w = x_{1}x_{2} ⋯ x_{n} such that each right factor v = x_{i} ⋯ x_{n} of w satisfies |v|_{j} ≥ |v|_{j+1} for each j, where |v|_{j} denotes the number of occurences of the letter j in v.

A Yamanouchi domino tableau is a domino tableau whose column reading is a Yamanouchi word. We denote by Yam_{2}(λ, μ) the set of Yamanouchi domino tableaux of shape λ and weight μ.

It follows from the results of [CL], Section 7, that the Schur expansions of the H-functions of level 2 are given by

H_{λ}^{(2)} = ∑_{μ} ∑_{T ∈Yam_{2}(2λ,μ)} q^{s(T)}s_{μ} .  (22)

On the other hand,

Q′_{λ} = ∑_{μ} ∑_{t∈Tab (μ,λ)} q^{c(t)}s_{μ} .  (23)

To prove Theorem 5.5, it is thus sufficient to exhibit an injection

η : Yam_{2}(2λ, μ) ⟶ Tab (μ, λ)

satisfying

c(η(T)) = s(T) .

To achieve this, we shall make use of a bijection described in [BV], and extended in [KLLT], which sends a domino tableau T ∈ Tab_{2}(α, μ) over the alphabet X = {1, …, n}, to an ordinary tableau t = ϕ(T) ∈ Tab (α, μ̄μ) over the alphabet X̄∪X = {n̄ < … < ¯1 < 1 < … < n}. The weight μ̄μ means that t contains μ_{i} occurences of i and of ī. The tableau ϕ(T) is invariant under Schützenberger’s involution Ω, and the spin of T can be recovered from t by the following procedure [KLLT2].

Let α = 2λ, β = α′, β_{odd} = (β_{1}, β_{3}, …) and β_{even} = (β_{2}, β_{4}, …). Then, there exists a unique factorisation t = τ_{1}τ_{2} in the plactic monoid Pl (X∪X̄), such that τ_{1} is a contretableau of shape α^{1} = (β_{even})′ and τ_{2} is a tableau of shape α^{2} = (β_{odd})′. The spin of T = ϕ^{−1}(t) is then equal to the number |τ_{1}|_{+} of positive letters in τ_{1}, which is also equal to the number |τ_{2}|_{−} of negative letters in τ_{2}. Moreover, τ_{2} = Ω(τ_{1}).

Example 6.1 With the following tableau T of shape (4, 4, 2, 2), one finds

1 1 1 2 2 3  ¯3 ¯2 ¯1 12 1 ¯1 ¯2¯11 32 t = T =  ϕ

By jeu-de-taquin, we find that in the plactic monoid

t = ¯1 1 ¯2 ¯1 ¯2 ¯3 3 2 1 2 ¯1 1 = τ_{1}τ_{2} .

The number of positive letters of τ_{1} and the number of negative letters of τ_{2} are both equal to 1, which is the spin of T.

This correspondence still works in the general case (α need not be of the form 2λ) and the invariant tableau associated to a domino tableau T admits a similar factorisation t = τ_{1}τ_{2}, but in general τ_{2} ≠ Ω(τ_{1}) and the formula for the spin is s(T) = \frac{1}{2}(|τ_{1}|_{+} + |τ_{2}|_{−}).

The map η : Yam_{2}(2λ, μ) ⟶ Tab (μ, λ) is given by the following algorithm: to compute η(T),

1. construct the invariant tableau t = ϕ(T)

2. apply the jeu-de-taquin algorithm to t to obtain the plactic factorization t = τ_{1}τ_{2}, and keep only τ_{2}.

3. Apply the evacuation algorithm to the negative letters of τ_{2}, keeping track of the successive stages. After all the negative letters have been evacuated, one is left with a Yamanouchi tableau τ in positive letters.

4. Complete the tableau τ to obtain the tableau t′ = η(T) using the following rule: suppose that at some stage of the evacuation, the box of τ_{2} which disappeared after the elimination of ī was in row j of τ_{2}. Then add a box numbered j to row i of τ.

Theorem 6.2 The above algorithm defines an injection

η : Yam_{2}(2λ, μ) ⟶ Tab (μ, λ)

satifying c ∘ η = s.

Corollary 6.3 H_{λ}^{(2)} ≤ Q′_{λ}

Example 6.4 Let T be the following Yamanouchi domino tableau, which is of shape 2λ = (6, 4, 4, 2, 2), of weight μ = (4, 3, 2) and has spin s(T) = 3

1 1 23 3 2 2 1 1 T =

Then,

ϕ(T) = 3 3 1 2 ¯1 ¯1 2 2 ¯2 ¯2 ¯1 1 ¯3 ¯3 ¯2 ¯1 1 1  ≡  ¯1 1 3 ¯1 2 ¯2 ¯1 ¯2 ¯3 3 2 1 2 ¯2 1 ¯3 ¯1 1

and the succesive stages of the evacuation process are

3 2 1 2 ¯2 1 ¯3 ¯1 1 ⟶ × ; 3 ; 2 2 ; 1 1 ; ¯2 ¯1 1 ⟶ 3 ; 2 × ; 1 2 ; ¯1 1 1 ⟶ × ; 3 ; 2 2 ; 1 1 1
ī = ¯3  ī = ¯2  ī = ¯1
j = 5  j = 3  j = 4

so that we find

η(T) = 3 5 2 2 3 1 1 1 4

a tableau of shape μ = (4, 3, 2), weight λ = (3, 2, 2, 1, 1) and charge c(t′) = 3.

7 The stable case

As the Q′-functions are known to verify all the conjectured properties of H-functions, the stable case of the conjectures will be a consequence of theorem 5.6. This result will be proved by means of Shimomura’s cell decomposition of unipotent varieties.

A tabloid t of shape ν = (ν_{1}, …, ν_{k}) can be identified with a k-tuple (w_{1}, …, w_{k}) of words, w_{i} being a row tableau of lenght ν_{i}. The Stanton-White correspondence ψ assciates to such a k-tuple of tableaux a k-ribbon tableau T = ψ(t). Thus, the cells of a unipotent variety F_{μ}^{u} (where u is of type ν) are labelled by k-ribbon tableaux of a special kind. The following theorem, which implies the stable case of the conjectures, shows that this labelling is natural from a geometrical point of view.

Theorem 7.1 The Stanton-White correspondence ψ sends a tabloid t ∈ L(ν, μ) onto a ribbon tableau T = ψ(t) whose cospin is equal to the dimension of the cell c_{t} of F_{μ}^{u} labelled by t, when one uses the modified indexation for which the dimension of c_{t} is e(t) (see Section 2). That is,

s̃(ψ(t)) = e(t) .

At this point, it is useful to observe, following [Te], that the e-statistic can be given a nonrecursive definition, as a kind of inversion number. Let t = (w_{1}, …, w_{k}) be a tabloid, identified with a k-tuple of row tableaux. Let y be the r-th letter of w_{i} and x be the r-th letter of w_{j}, and suppose that x < y. Then, the pair (y, x) is said to be an e-inversion if either

(a) i < j

or

(b) i > j and there s on the right of x in w_{j} a letter u < y

Then e(t) is equal to the number of inversions (y, x) in t.

Example 7.2 Let t ∈ L((2, 3, 2, 1), (2, 3, 1, 1, 1)) be the following tabloid (the number under a letter y is the number of e-inversions of the form (y, x)):

t = \binom{2 3 , 1 1 2 , 4 5 , 2}{1 1  0 0 0  3 1  1}

so that e(t) = 7. Its image under the SW-correspondence is the 4-ribbon tableau

1 2 2 4 1 3 5 2

whose cospin is equal to 7.

References

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