Refined Gelfand models for wreath products

TL;DR

This paper refines Gelfand models for wreath products of cyclic groups with symmetric groups, providing a combinatorial description of their irreducible decomposition aligned with the Robinson-Schensted correspondence.

Contribution

It offers a simple combinatorial description of the irreducible decomposition of Gelfand models for wreath products, extending previous models to a broader class of groups.

Findings

Decomposition compatible with Robinson-Schensted correspondence

Explicit combinatorial description of submodules

Enhanced understanding of involutory reflection groups

Abstract

In [F. Caselli, Involutory reflection groups and their models, J. Algebra 24 (2010), 370--393] it is constructed a uniform Gelfand model for all non-exceptional irreducible complex reflection groups which are involutory. This model can be naturally decomposed into the direct sum of submodules indexed by symmetric conjugacy classes, and in this paper we present a simple combinatorial description of the irreducible decomposition of these submodules if the group is the wreath product of a cyclic group with a symmetric group. This is attained by showing that such decomposition is compatible with the generalized Robinson-Schensted correspondence for these groups.