Some nonsimple modules for centralizer algebras of the symmetric group

Dodge, Craig; Ellers, Harald; Nakada, Yukihide; Pohland, Kelly

doi:10.2140/involve.2016.9.877

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ISSN: 1944-4184 (e-only)

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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

DOI: 10.2140/involve.2016.9.877

Abstract

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

Vol. 9, No. 5, 2016