Some nonsimple modules for centralizer algebras of the symmetric group

Abstract

James classified the simple modules over the group algebra $k{\Sigma }_n$ using modules denoted $D^{\lambda }$, where $\lambda$ is a partition of $n$. In particular, he showed that $D^{\lambda }$ is simple or zero for every partition $\lambda$ and, furthermore, that for every simple $k{\Sigma }_n$-module $S$ there exists a partition $\lambda$ such that $D^{\lambda }\cong S$. This paper is an extension of a paper of Dodge and Ellers in which they studied analogous modules ${\mathcal{D}}^{\left(\lambda ,\mu \right)}$ over the centralizer algebra $k{\Sigma }_n^{{\Sigma }_l}$, where $\lambda$ is a partition of $n$ and $\mu$ a partition of $l$. For every positive prime $p$ we find counterexamples to their conjecture that the $k{\Sigma }_n^{{\Sigma }_l}$-modules ${\mathcal{D}}^{\left(\lambda ,\mu \right)}$ are always simple or zero, where $k$ is a field of characteristic $p$. We also study the relationship between ${\mathcal{D}}^{\left(\lambda ,\mu \right)}$ and ${Hom}_{k{\Sigma }_l}\left(D^{\mu },{res}_{{\Sigma }_l}^{{\Sigma }_n}{D}^{\lambda }\right)$ in special cases.