Unisingular Specht Modules

Abstract

Let $G$ be a finite group and $ρ:G \to \GL(V)$ a finite dimensional representation of $G$. We say that $\ddot{I}$ is unisingular if $\det (1-\ddot{I}(g))=0$ 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.