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GREEN POLYNOMIALS AND HALL-LITTLEWOOD

FUNCTIONS AT ROOTS OF UNITY

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

Abstract. — We give some applications of our recent work [LLT] about Hall-Littlewood functions at roots of unity. In particular, we prove the two conjectures of N. Sultana [Su] on specializations of Green polynomials, and we generalize results of Kraskiewicz-Weyman and Molev-Tsalenko concerning characters of the symmetric group induced by maximal cyclic subgroups. We also give some new congruences for Kostka-Foulkes polynomials modulo cyclotomic polynomials.

1. Introduction

Hall-Littlewood functions are symmetric functions of a set of variables A depending on a parameter q (see [Mcd]). When q = −1, these 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 A.O. Morris [Mo], in connection with problems in the modular representation theory of the symmetric group. Recently, new results on this question have been obtained [Su], [MS], [LLT]. In [Su] and [MS], one finds some conjectures about Green polynomials (which are q-analogs of the characters of symmetric groups). These conjectures are proved in section 3.

Section 4 is concerned with some combinatorial properties of certain symmetric functions discovered by Foulkes [Fo]. These functions correspond to characters of the symmetric group induced by irreducible representations of a maximal cylic subgroup. 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. We give here a generalization of this result, in which the coefficients of certain plethysms with the preceding functions 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.

† L.I.T.P., Institut Blaise Pascal, Université Paris 7, 2 Place Jussieu, 75251 Paris Cedex 05, France.

‡ Institut Gaspard Monge, Université de Marne-la-Vallée, 2 Allée Jean Renoir, 93160 Noisy-Le-Grand, France.

In section 5, we derive some congruences for Kostka-Foulkes polynomials modulo cyclotomic polynomials. Such congruences were derived in [LLT] for certain columns of the q-Kostka matrix. Those presented here involve rows of the same matrix.

All the results of this paper are consequences of two theorems from [LLT], which we recall in section 2.

2. Notations and background

Our notations for symmetric functions are as in [LLT]. Partitions are for us weakly increasing vectors of nonnegative integers I = (i_{1} ≤ i_{2} ≤ … ≤ i_{m}), Schur functions of an alphabet A are denoted by S_{I}(A), products of power-sums ψ_{i_{1}}ψ_{i_{2}} ⋯ ψ_{i_{m}} by ψ^{I}(A), and Hall-Littlewood functions by P_{I}(A; q), Q_{I}(A; q), as in [Mcd]. The modified Hall-Littlewood functions are the Q′_{I}(A; q) = Q_{I}(A/(1 − q); q) (in the sense of λ-rings), that is, the Q′_{I} form the adjoint basis of the P_{I} for the standard scalar product ( , ) (for which the Schur functions form an orthonormal basis). The Hall-Littlewood scalar product is denoted by ⟨., .⟩_{q}.

The Green polynomials (see [Mcd]) X_{J}^{I}(q) are defined by X_{J}^{I}(q) = ⟨Q_{I}, ψ^{J}⟩_{q} = (Q′_{I}, ψ^{J}). Given two partitions I and J, we denote by I ∨ J the partition obtained by reordering the concatenation of I and J, e.g. (122) ∨ (1125) = (1112225). We write I^{k} for I ∨ I ∨ ⋯ ∨ I (k factors). If I = (i_{1}, …, i_{r}) we set kI = (ki_{1}, …, ki_{r}). Given a symmetric function F, we denote by D_{F} the adjoint of the linear operator G ↦ F G (multiplication by F). We shall make use of the following two results, which were established in [LLT]:

Theorem 2.1. — Let I = (1^{m_{1}}2^{m_{2}} … n^{m_{n}}) be a partition, write m_{i} = kp_{i} + r_{i} with 0 ≤ r_{i} < k, and H = (1^{r_{1}}2^{r_{2}} … n^{r_{n}}). Then, ζ being a primitive k-th root of unity,

Q′_{I}(A; ζ) = Q′_{H}(A; ζ) ∏_{i≥1}[Q′_{(i^{k})}(A; ζ)]^{p_{i}} .

Theorem 2.2. — Let ψ_{k}(S_{n}) denote the plethysm of the complete function S_{n} by the power-sum ψ_{k}, which is defined by the generating series ∑_{n} z^{n}ψ_{k}(S_{n}(A)) = ∏_{a∈A}(1−za^{k})^{−1}. Then, if ζ is as above a primitive k-th

root of unity, one has

Q′_{(n^{k})}(A; ζ) = (−1)^{(k−1)n}ψ_{k}(S_{n}(A)).

3. Green polynomials at roots of unity

Let ζ ∈ ℂ be a primitive k-th root of unity. The following two results have been conjectured in [Su] (see also [MS]):

Theorem 3.1. — Let I = (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 J = (j_{1}, …, j_{r}) be another partition,

with at least one part j_{h} ≡ 0 (mod k). Then,

X_{J}^{I}(ζ) ≡ 0 (mod k) .

Theorem 3.2. — Let I = (1^{m_{1}} … i^{m_{i}} … n^{m_{n}}), where for some i, m_{i} ≥ 1, |I| = n, I′ = (1^{m_{1}} … i^{m_{i}−1} … n^{m_{n}}) and suppose n = kr. Then,

X_{I}^{(r^{k})}(ζ) = (−1)^{(k−1)j}k X_{I′}^{(r−j)^{k}}(ζ)

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

Proof of Theorem 3.1: Put I = (j^{k}) ∨ I′ and J = (k · H) ∨ J′, where no part of J′ is multiple of k. Then, all HL-functions being understood as spe-

cialized at q = ζ,  X_{J}^{I}(q) = (ψ^{J}, Q′_{I}) = (ψ^{kH}ψ^{J′}, Q′_{(j^{k})}Q′_{I′})  (by Theorem 2.1)
= ±(ψ^{kH}ψ^{J′}, ψ_{k}(S_{j})Q′_{I′}) (by Theorem 2.2 )

= ±(D_{ψ_{k}(S_{j})}(ψ^{kH}ψ^{J′}) , Q′_{I′})

so that we just have to check that the coefficients of the expansion of D_{ψ_{k}(S_{j})}(ψ^{kH}ψ^{J′}) in the basis (ψ^{K}) are in kℤ. But these coefficients can be explicitely computed: one has

D_{ψ_{k}(S_{j})}(ψ^{kH}ψ^{J′}) = ψ^{J′}D_{ψ_{k}(S_{j})}ψ^{kH}

and

D_{ψ_{k}(S_{j})}ψ^{kH} = ∑_{|M|=j} \frac{1}{(ψ^{M}, ψ^{M})}D_{ψ^{kM}} ψ^{kH}

with (D_{ψ^{kM}} ψ^{kH}, ψ^{kN}) = (ψ^{kH}, ψ^{k·(M∨N)}) ≠ 0 only when H = M ∨ N.

Now, let H = (1^{a_{1}}2^{a_{2}} …), M = (1^{b_{1}}2^{b_{2}} …) and N = (1^{c_{1}}2^{c_{2}} …). We have therefore

D_{ψ^{kM}} ψ^{kH} = (ψ^{kH}, ψ^{kH})\frac{ψ^{kN}}{(ψ^{kN}, ψ^{kN})}

= ∏_{i} \frac{(ki)^{a_{i}}a_{i}!}{(ki)^{c_{i}}c_{i}!}ψ^{kN}

whence

\frac{1}{(ψ^{M}, ψ^{M})}D_{ψ^{kM}} ψ^{kH} = k^{Σa_{i}−Σc_{i}} ∏_{i} \frac{a_{i}!}{b_{i}!c_{i}!}ψ^{kN}

= k^{ℓ(M)} ∏_{i} \binom{a_{i}}{b_{i}}ψ^{kN}

and this number is congruent to 0 since ℓ(M) > 0.

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

X_{224}^{11114}(i) = (Q′_{1111}Q′_{4}, ψ^{4}ψ^{22}) = (−1)^{3}(ψ_{4}Q′_{4}, ψ^{4}ψ^{22})
(Q′_{4}, D_{ψ_{4}}(ψ^{4}ψ^{22})) = −(Q′_{4}, 4ψ^{22}) = −4(S_{4}, ψ^{22}) = −4 .

To prove Theorem 3.2, we need the following lemma, which is easily established by induction [CT]:

Lemma 3.3. — Let F ∘ G denote (as in [Mcd]) the plethysm of the two symmetric functions F and G. Then,

D_{ψ_{m}}(F ∘ G) = ∑_{d|m}[(D_{ψ_{d}}F ) ∘ G] · [ψ_{d} ∘ (D_{ψ_{m/d}}G)] .

Proof of Theorem 3.2: The HL-functions being as above specialized at q = ζ, we have

X_{I}^{(r^{k})}(ζ) = (ψ^{I}, Q′_{(r^{k})}) = (−1)^{(k−1)r}(ψ^{I}, ψ_{k}(S_{r}))

(by Theorem 2.2 )

= (−1)^{(k−1)r}(ψ^{I′}ψ_{i}, ψ_{k}(S_{r})) = (−1)^{(k−1)r}(ψ^{I′}, D_{ψ_{i}}[ψ_{k} ∘ S_{r}])

where by Lemma 3.3,

D_{ψ_{i}}[ψ_{k} ∘ S_{r}] = ∑_{d|i}[(D_{ψ_{d}}ψ_{k}) ∘ S_{r}] · ψ_{d} ∘ (D_{ψ_{i/d}}S_{r})

= k · ψ_{k}(D_{ψ_{j}}S_{r}) = kψ_{k}(S_{r−j}) = (−1)^{(k−1)(r−j)}kQ′_{(r−j)^{k}}

if i = kj and = 0 otherwise (by Theorem 2.2 ), whence X_{I}^{(r^{k})}(ζ) =

(−1)^{(k−1)j}(ψ^{I}, kQ′_{(r−j)^{k}}) = (−1)^{(k−1)j}kX_{I′}^{(r−j)^{k}}(ζ), as required.

More generally, when I = H^{k} = H ∨ H ∨ ⋯ ∨ H (k factors), X_{J}^{I}(ζ) can be expressed by means of the value of a permutation character of 𝔖_{n}:

X_{J}^{H^{k}}(ζ) = (−1)^{(k−1)|H|}(ψ^{J}, ψ_{k}(S^{H})) = (−1)^{(k−1)|H|}(ϕ_{k}(ψ^{J}), S^{H}) ,

(where ϕ_{k} is the adjoint of the linear operator F ↦ ψ_{k} ∘ F, and S^{H} = S_{h_{1}}S_{h_{2}} ⋯ S_{h_{r}}) so that:

Theorem 3.4. — X_{J}^{H^{k}}(ζ) = 0 when J is not of the form kM for some partition M, and

X_{kM}^{H^{k}}(ζ) = (−1)^{(k−1)|H|}k^{ℓ(M)}(ψ^{M}, S^{H}) .

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

Examples. — (i) With k = 4 (ζ = i), X_{224}^{2222}(i) = (ψ_{4}(S_{2}), ψ^{224}) = 0 and X_{44}^{24}(i) = (S_{2}, ϕ_{4}(ψ^{44})) = (S_{2}, 16ψ^{11}) = 16.

(ii) With k = 3 (ζ = e^{2πi/3}), X_{339}^{222333}(ζ) = +3^{3}(ψ^{113}, S^{23}) = 3^{3}(ψ^{11}, S_{2}D_{ψ_{3}}S_{3}) = 27.

4. Charge congruences, harmonic polynomials and cohomology of flag manifolds

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 [De] for a simpler proof), provides a generalization of the Moebius inversion formula:

Lemma 4.1. — 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

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

(ii) 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)ψ_{d}^{n/d} .

These functions were first obtained by Foulkes as Frobenius characteristics of the representations of the symmetric group induced by irreducible representations of a maximal cyclic subgroup generated by an n-cycle [Fo]. A combinatorial interpretation of the multiplicity (S_{I}, ℓ_{n}^{(k)}) has been discovered by Kraskiewicz and Weyman [KW]. A proof of this result by means of Cohen’s theorem (Lemma 4.1) was subsequently found by Désarménien [De]. Adapting the argument of [De] to our case, we arrive at the following generalization of the Kraskiewicz-Weyman theorem:

Theorem 4.2. — Let Λ_{i} be the i-th elementary symmetric function, and for H = (h_{1}, …, h_{m}), Λ^{H} = Λ_{h_{1}} ⋯ Λ_{h_{m}}. Then, the multiplicity (S_{I}, ℓ_{k}^{(r)} ∘ Λ^{H}) of the Schur function S_{I} in the plethysm ℓ_{k}^{(r)} ∘ Λ^{H} is equal to the number of Young tableaux of shape I ˜ (conjugate partition) and evaluation H^{k} whose charge is congruent to r modulo k.

Proof: Let |I| = n, |H| = m and d | k. If P (q) = ∑_{i=0}^{k−1} a_{i}q^{i} is the residue

modulo 1−q^{k} of the Kostka polynomial K_{I,H^{k}}(q), one has, for ζ a primitive d-th root of unity,

P (ζ) = K_{I,H^{k}}(ζ) = (−1)^{(d−1)mk/d}(ψ_{d}^{k/d}(S^{H}), S_{I})

(by Theorem 2.2)

= (−1)^{n}(ψ_{d}^{k/d} ∘ ((−1)^{m}S^{H}), S_{I}) = ((−1)^{n}ψ_{d}^{k/d} ∘ Λ^{H}(−A), S_{I}(A))

= (ψ_{d}^{k/d}(Λ^{H}), S_{I} ˜ ) = r_{d} ∈ ℤ .

Hence, by Lemma 4.1 (i),

a_{i} = \frac{1}{k} ∑_{d|k} c(i, d)(ψ_{d}^{k/d}(Λ^{H}), S_{I} ˜ ) = (ℓ_{k}^{(i)}(Λ^{H}), S_{I} ˜ ) .

Example. — With k = 4, r = 2 and H = (2),

ℓ_{4}^{(2)}(Λ^{2}) = S_{134} + S_{224} + S_{11114} + 2S_{1133} + 2S_{1223}
+2S_{11123} + S_{2222} + S_{11222} + 2S_{111122} + S_{1111112} .

To compute the coefficient (S_{11123}, ℓ_{4}^{(2)}(Λ^{2})) = 2, we have to find the number of tableaux of shape (11123) ˜ = (125), evaluation (2222) 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_{237}, ℓ_{4}^{(2)}(Λ^{12})) = 5 is the number of tableaux with shape (1111233), evaluation (11112222) and charge ≡ 2 (mod 4).

A related result, due to Molev and Tsalenko [MT], gives an interpretation of ℓ_{n}^{(k)} in terms of 𝔖_{n}-harmonic polynomials: let H_{n}^{k} be the Frobenius characteristic of the homogeneous component H_{n}^{k} of degree k of the module H_{n} of 𝔖_{n}-harmonic polynomials. Then,

ℓ_{n}^{(k)} = ∑_{i≡k (mod n)} H_{n}^{i} .

More generally, the plethysm ℓ_{k}^{(j)}(Λ^{H}) can be expressed as a positive sum of Frobenius characteristics of modules of harmonic polynomials. To each partition J of n, it is possible to associate a certain submodule H_{J} of H_{n} which has been defined in [La] and called an affine Specht module. This module is isomorphic, as a graded 𝔖_{n}-module, to the rational cohomology H^{∗}(F_{J}) of the variety of flags fixed by an unipotent of Jordan canonical form corresponding to the partition J. The action of the symmetric group on H^{∗}(F_{J}) is defined in [HS], and the Frobenius characteristic H_{J}^{j} = ch(H^{2j}(F_{J})) = ch(H_{J}^{j}) is given by the equation

∑_{j} q^{j}H_{J}^{j} = ∑_{I} K̃_{I ˜ ,J}(q)S_{I}

where the K̃_{I,J}(q) are the cocharge polynomials (i.e. the reciprocals of the Kostka polynomials, see also [Mcd], ex. 9 p. 136).

Now, suppose J = H^{k}, and let K̃_{I ˜ ,J}(q) ≡ ∑_{j=0}^{k−1} a_{j}q^{j} (mod 1 − q^{k}). Since

(S_{I}, H_{J}^{i}) is equal to the coefficient of q^{j} in K̃_{I ˜ ,J}(q), we have

(S_{I}, ∑_{i≡j (mod k)} H_{J}^{i}) = a_{j} .

On the other hand, K̃_{I ˜ ,J}(q) ≡ K_{I ˜ ,J}(q) (mod 1 − q^{k}) whence a_{j} = (ℓ_{k}^{(j)}(Λ^{H}), S_{I}). To summarize:

Theorem 4.3. — Let J = H^{k}. Then,

ℓ_{k}^{(j)}(Λ^{H}) = ∑_{i≡j (mod k)} H_{J}^{i} .

The theorem of Molev and Tsalenko corresponds to the case H = (1) i.e. to the case of the full flag manifold.

Example. — The character of the preceding example decomposes into three parts

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

the three components being

H_{2222}^{2} = S_{111122} + S_{1111112} ,

H_{2222}^{6} = S_{1133} + 2S_{1223} + S_{2222} + S_{11114} + 2S_{11123} + S_{11222} + S_{111122}

and H_{2222}^{10} = S_{134} + S_{224} + S_{1133}.

5. Further congruences for Kostka polynomials

We have given in [LLT] some congruences modulo cyclotomic polynomials for certain columns of the q-Kostka matrix. We will now present other congruences for certain rows of the same matrix.

Let ρ_{n} be the staircase partition (12 … n). It is well-known that for k = 2 (ζ = −1), one has S_{ρ_{n}} = P_{ρ_{n}}. More generally, it can be shown that if ζ is a primitive k-th root of unity, P_{ρ_{n}^{k−1}}(A; ζ) = S_{ρ_{n}^{k−1}}(A). This is a consequence of the following result:

Theorem 5.1. — Let ζ be a primitive k-th root of 1, and I be a partition such that ℓ(I ˜ ) ≤ n. Then,

S_{I^{k}∨ρ_{n}^{k−1}}(A) = ∑_{J} K_{IJ} P_{J^{k}∨ρ_{n}^{k−1}}(A; ζ)

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

K_{I^{k}∨ρ_{n}^{k−1}, H}(q) ≡ \{ \begin{matrix}K_{IJ} & if & H = J^{k} ∨ ρ_{n}^{k−1} \\ 0 & otherwise. \end{matrix}

This theorem is itself a direct consequence of the following three lemmas. Details about k-cores and rim hook tableaux can be found in [KSW] and [SW].

Lemma 5.2. — Let I be a partition such that ℓ(I ˜ ) ≤ n.

a) The k-core of the partition I^{k} ∨ ρ_{n}^{k−1} is ρ_{n}^{k−1}.

b) The only k-rim hooks which are removable from I^{k}∨ρ_{n}^{k−1} are those of shape (1^{k}). After removal of such a rim hook, one gets a partition I^{′k}∨ρ_{n}^{k−1} with ℓ(I′ ˜ ) ≤ n.

Lemma 5.3. — If (Q′_{H}, S_{I^{k}∨ρ_{n}^{k−1}}) ≠ 0, then H = J^{k} ∨ ρ_{n}^{k−1} for some partition J.

Proof: If (Q′_{H}, S_{I^{k}∨ρ_{n}^{k−1}}) ≠ 0 then ℓ(H) ≥ ℓ(I^{k} ∨ ρ_{n}^{k−1}). This forces H to have a part j with multiplicity ≥ k. Set H = (j^{k}) ∨ H′. Then,

(Q′_{H}, S_{I^{k}∨ρ_{n}^{k−1}}) = (−1)^{(k−1)j}(Q′_{H′}, D_{ψ_{k}(S_{j})}S_{I^{k}∨ρ_{n}^{k−1}})

(by Theorems 2.1 and 2.2)

= (−1)^{(k−1)j}(−1)^{(k−1)j} ∑_{I′} α_{I′}(Q′_{H′}, S_{I′∨ρ_{n}^{k−1}}) = ∑_{I′} α_{I′}(Q′_{H′}, S_{I^{′k}∨ρ_{n}^{k−1}})

(by Lemma 5.2) where the summation is over partitions I′ of weight |I| − kj, and the α_{I′} are certain nonnegative integers. By induction, we obtain after a finite number of steps

(Q′_{H}, S_{I^{k}∨ρ_{n}^{k−1}}) = ∑_{H′′} α_{H′′}(Q′_{H′′}, S_{ρ_{n}^{k−1}})

where α_{H′′} ≥ 0. For this sum to be non zero, it is necessary that for some H′′, (Q′_{H′′}, S_{ρ_{n}^{k−1}}) ≠ 0. This implies that ℓ(H′′) ≥ ℓ(ρ_{n}^{k−1}) and that no multiplicity of H′′ be ≥ k. The only possibility is H′′ = ρ_{n}^{k−1}, whence H = H′′ ∨ J^{k} = ρ_{n}^{k−1} ∨ J^{k}.

Lemma 5.4. — For |I| = |J| and ℓ(I ˜ ) ≤ n,

D_{ψ_{k}(S^{J})}S_{I^{k}∨ρ_{n}^{k−1}} = (−1)^{(k−1)|J|} K_{IJ}S_{ρ_{n}^{k−1}} .

Proof: Since |I| = |J| and the k-core of I^{k} ∨ ρ_{n}^{k−1} is ρ_{n}^{k−1}, we see that

D_{ψ_{k}(S^{J})}S_{I^{k}∨ρ_{n}^{k−1}} = β_{IJ} · S_{ρ_{n}^{k−1}}

where β_{IJ} is, up to sign, the number of k-rim hook tableaux of shape I^{k} ∨ ρ_{n}^{k−1}/ρ_{n}^{k−1} and evaluation J. From lemma 5.2, it results that the rim hooks of these tableaux are all of shape (1^{k}), so that β_{IJ} = (−1)^{(k−1)|J|}K_{IJ}.

Proof of Theorem 5.1:

S_{I^{k}∨ρ_{n}^{k−1}} = ∑_{H}(Q′_{H}, S_{I^{k}∨ρ_{n}^{k−1}}) P_{H}

= ∑_{J}(Q′_{J^{k}∨ρ_{n}^{k−1}}, S_{I^{k}∨ρ_{n}^{k−1}})P_{J^{k}∨ρ_{n}^{k−1}}

(by Lemma 5.3)

= ∑_{J}(−1)^{(k−1)|J|}(Q′_{ρ_{n}^{k−1}}, D_{ψ_{k}(S^{J})}S_{I^{k}∨ρ_{n}^{k−1}})P_{J^{k}∨ρ_{n}^{k−1}}

(by Theorems 2. and 2.2)

= ∑_{J} K_{IJ}P_{J^{k}∨ρ_{n}^{k−1}}

(by Lemma 5.4).

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

K_{112222233333,111112222222233}(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_{23,122} = 2 (mod 1 + q + q^{2}) .

References

[CT] C. Carré and J.-Y. Thibon, Some formulas for plethysm coefficients, L.I.T.P. Internal report 91.63 (1991).

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

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

[Fo] 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.

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

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

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

[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.

[La] A. Lascoux, Cyclic permutations on words, tableaux and harmonic polynomials, Proc. of the Hyderabad Conference on Algebraic groups, Manoj Prakashan, Madras, 1991, 323-347.

[LLT] 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, t. 316, ser. I (1993), 1-6.

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

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

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

[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.

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

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