On Sch\"utzenberger modules of the cactus group

TL;DR

This paper explores how the cactus group acts on Specht modules associated with Young tableaux, revealing specific decomposition patterns for hook shapes and connecting to Kostka coefficients.

Contribution

It introduces Sch"utzenberger modules for the cactus group and characterizes their decomposition for hook shapes, extending the understanding of group actions on Specht modules.

Findings

Cactus group action on hook-shaped modules factors through symmetric group S_{n-1}.

Decomposition multiplicities are given by Kostka coefficients.

The proof uses results from Berenstein-Kirillov and Chmutov-Glick-Pylyavskyy.

Abstract

The cactus group acts on the set of standard Young tableau of a given shape by (partial) Sch\"utzenberger involutions. It is natural to extend this action to the corresponding Specht module by identifying standard Young tableau with the Kazhdan-Lusztig basis. We term these representations of the cactus group "Sch\"utzenberger modules", denoted $S^\lambda_{\mathsf{Sch}}$, and in this paper we investigate their decomposition into irreducible components. We prove that when $\lambda$ is a hook shape, the cactus group action on $S^\lambda_{\mathsf{Sch}}$ factors through $S_{n-1}$ and the resulting multiplicities are given by Kostka coefficients. Our proof relies on results of Berenstein and Kirillov and Chmutov, Glick, and Pylyavskyy.