Maciej Dołęga
Title: Multigraded action of the Hecke algebra on the coinvariant algebra
Abstract:
A classical result of Chevalley states that the coinvariant algebra has a structure of an $S_n$-module
isomorphic to the regular representation. It is very interesting to decompose each homogeneous
component of the coinvariant algebra into irreducibles, which was done independently by Stanley
and Lusztig (unpublished). This story has two natural refinements. First, by decomposing homogenous
part into submodules generated by monomials and studying their decomposition into irreducibles,
which was done by Adin, Brenti and Roichman. Second, by introducing $q$--quantization through
the Hecke algebra $H_n$ of the permutation group, where the explicit decomposition into irreducibles
was found by Adin, Postnikov, and Roichman. We are going to refine these results by providing a
formula treating both cases simultaneously, thus we decompose each $H_n$--submodule generated
by monomials into irreducible components. As an application we introduce the $q$--deformation of
Foulkes characters answering a question of Stanley.