Gelfand models and Robinson-Schensted correspondence

TL;DR

This paper explores the connection between Gelfand models for involutory complex reflection groups and the projective Robinson-Schensted correspondence, revealing insights into the structure and split representations of these groups.

Contribution

It establishes a general link between the irreducible decomposition of Gelfand model submodules and the projective Robinson-Schensted correspondence for involutory reflection groups.

Findings

Decomposition of Gelfand models aligns with the projective Robinson-Schensted correspondence.

Explicit description of split representations for involutory reflection groups.

Unified approach for non-exceptional complex reflection groups.

Abstract

In [F. Caselli, Involutory reflection groups and their models, J. Algebra 24 (2010), 370--393] there is constructed a uniform Gelfand model for all non-exceptional irreducible complex reflection groups which are involutory. Such model can be naturally decomposed into the direct sum of submodules indexed by $S_n$-conjugacy classes, and we present here a general result that relates the irreducible decomposition of these submodules with the projective Robinson-Schensted correspondence. This description also reflects in a very explicit way the existence of split representations for these groups.