#### Vol. 9, No. 5, 2016

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Some nonsimple modules for centralizer algebras of the symmetric group

### Craig Dodge, Harald Ellers, Yukihide Nakada and Kelly Pohland

Vol. 9 (2016), No. 5, 877–898
##### 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 ${\mathsc{D}}^{\left(\lambda ,\mu \right)}$ over the centralizer algebra $k{\Sigma }_{n}^{{\Sigma }_{l}}\phantom{\rule{0.3em}{0ex}}$, 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 ${\mathsc{D}}^{\left(\lambda ,\mu \right)}$ are always simple or zero, where $k$ is a field of characteristic $p$. We also study the relationship between ${\mathsc{D}}^{\left(\lambda ,\mu \right)}$ and ${Hom}_{k{\Sigma }_{l}}\left({D}^{\mu },{res}_{{\Sigma }_{l}}^{{\Sigma }_{n}}{D}^{\lambda }\right)$ in special cases.

##### Keywords
centralizer algebras, symmetric groups, modular representations
##### Mathematical Subject Classification 2010
Primary: 20C05, 20C20
##### Milestones
Received: 16 October 2015
Revised: 15 February 2016
Accepted: 13 May 2016
Published: 25 August 2016

Communicated by Kenneth S. Berenhaut
##### Authors
 Craig Dodge Department of Mathematics Allegheny College 520 North Main St. Meadville, PA 16335 United States Harald Ellers Department of Mathematics Allegheny College 520 North Main St. Meadville, PA 16335 United States Yukihide Nakada Department of Mathematics Allegheny College 520 North Main St. Meadville, PA 16335 United States Kelly Pohland Department of Mathematics Allegheny College 520 North Main St. Meadville, PA 16335 United States