Unisingular Specht Modules

TL;DR

This paper investigates unisingular representations of symmetric groups, proving that certain Specht modules always have the property that their associated determinants vanish for all group elements, and raises new questions for future research.

Contribution

It establishes that specific families of Specht modules for symmetric groups are always unisingular, expanding understanding of their representation-theoretic properties.

Findings

Certain Specht modules are always unisingular for symmetric groups

The paper raises new questions for future research in representation theory

Builds on previous work to deepen understanding of unisingular representations

Abstract

Let $G$ be a finite group and $\rho:G \to \GL(V)$ a finite dimensional representation of $G$. We say that $\rho$ is unisingular if $\det(1-\rho(g)) = 0$ for all $g \in G$. Building on previous work in \cite{cullinan}, we consider the symmetric groups $S_n$ and prove that certain families of Specht modules are always unisingular as well as raise new questions for future study.