Indicators of Bismash Products from Exact Symmetric Group Factorizations

TL;DR

This paper investigates the Frobenius-Schur indicators of irreducible representations of bismash products derived from exact factorizations of symmetric groups, providing new theoretical insights and classifications.

Contribution

It proves that all irreducible representations of certain bismash products have indicators +1 or 0 and describes non-self-dual simple modules, advancing understanding of these algebraic structures.

Findings

All irreducible representations have indicators +1 or 0

Characterization of non-self-dual simple modules

New results and conjectures for bismash products from symmetric group factorizations

Abstract

We prove that all irreducible representations of the bismash product $H = \mathbb{k} ^G \# \mathbb{k} S_{n-k}$ have Frobenius-Schur indicators +1 or 0 where $\mathbb{k}$ is an algebraically closed field and $S_n = S_{n-k}\cdot G$ is an exact factorization. Moreover, we have an description of those simple modules which are not self-dual. We prove some new results for bismash products in general and make conjectures about bismash products that arise from other exact factorizations of $S_n$.