A formula of Lascoux-Leclerc-Thibon and representations of symmetric groups

Abstract

We consider Green polynomials at roots of unity, corresponding to partitions which we call l-partitions. We obtain a combinatorial formula for Green polynomials corresponding to l-partitions at primitive lth roots of unity. The formula is rephrased in terms of representation theory of the symmetric group.

Tabloids and weighted sums of characters of certain modules of the symmetric groups

“…We also introduce combinatorial objects called marked (ρ, l)-tabloids to describe weighted sums of characters of M µ (k; l). When n = 1, the number of marked (ρ, (l))-tabloids coincides with the right hand side of the explicit formula (3.1) of Green polynomials in [5]. Our main result is the description of a weighted sum k∈Z/lZ ζ jk l Char (M µ (k; l)) (σ) of characters of M µ (k; l) as the number of marked (ρ, γ)-tabloids on µ for the primitive l-th root ζ l of unity and σ ∈ S m of cycle type ρ in Section 3.…”

“…Remark 2.2. For an l-partition µ of m, our module M (µ) (k; (l)) gives a realization of the S m -module Ind Sm Hµ(l) Z µ (k; l) in Morita-Nakajima [5]. For n-tuple { l h } of integers, M µ (k; l) is a realization of the induced module Ind Sm 1 +···+mn…”

“…We call a pair (Y, c) a marked (ρ, l)-tabloid on an l-partition µ of m if the following are satisfied: Remark 2.8. It follows from a direct calculation that the number of marked (ρ, (l))tabloids on an (l)-partition (µ) equals the right hand side of the equation (3.1) in Morita-Nakajima [5].…”

“…In Section 2, we introduce S m -modules M µ and their quotient modules M µ (k; l) for some n-tuple µ of Young diagrams. When n = 1, this module M (µ) (k; (l)) is a realization of Ind Sm Hµ(l) Z µ (k; l) in [5]. We also introduce combinatorial objects called marked (ρ, l)-tabloids to describe weighted sums of characters of M µ (k; l).…”

Garsia–Haiman modules for hook partitions and Green polynomials with two variables

“…In [Mt2], the author considers a certain combinatorial property of the Springer modules, and gives an interpretation for the property in terms of representation theory of the symmetric group. For related works, see [BLM,KW,Mt1,MN1,MN2,Sh,St]. The aim of this paper is to show a similar result for the Garsia-Haiman module D μ in the case where the corresponding partition μ is a hook.…”