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 kΣn using modules denoted Dλ, where λ is a partition of n. In particular, he showed that Dλ is simple or zero for every partition λ and, furthermore, that for every simple kΣn-module S there exists a partition λ such that DλS. 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 kΣnΣl, 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 kΣnΣl-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 HomkΣl(Dμ,resΣlΣnDλ) in special cases.

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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