Branching rules for modular representations of symmetric groups. IV
type: Item
instance of: scholarly article
title: Branching rules for modular representations of symmetric groups. IV
zbMATH Open document ID: 0931.20014
published in: Journal of Algebra
publication date: 1999-11-10T00:00:00Z
review text: [For part III see J. Lond. Math. Soc., II. Ser. 54, No. 1, 25-38 (1996; Zbl 0854.20014).] Let \(\Sigma_n\) be the symmetric group on \(n\) letters, and \(D\) be an irreducible representation of \(\Sigma_n\) over a ground field \(F\) of characteristic \(p>0\). The main results of this paper are: (1) We describe the algebra \(\text{End}_{\Sigma_{n-1}}(D \downarrow_{\Sigma_{n-1}})\); (2) we obtain a criterion for the semisimplicity of the induced module \(D \uparrow^{\Sigma_{n+1}}\), and in the case of semisimplicity describe its composition factors. We note that for characteristic \(p=2\) the criterion for the semisimplicity we get was conjectured by Benson. Thus we generalize to an arbitrary \(p\) and prove the last unproved conjecture made by Benson at the Arcata Conference. Actually, (2) easily follows from (1) using a Mackey type theorem together with our previous results on \(D \uparrow^{\Sigma_{n+1}}\).
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 05E10
Mathematics Subject Classification ID: 20C20
zbMATH DE Number: 1214114
zbMATH Keywords: symmetric groups
zbMATH Keywords: irreducible representations
zbMATH Keywords: induced modules
zbMATH Keywords: semisimplicity
zbMATH Keywords: Mackey type theorems
author: Alexander S. Kleshchev
MaRDI profile type: Publication
full work available at URL: jabr.1997.7302
OpenAlex ID: W4213155145
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DOI: 10.1006/JABR.1997.7302
description: scientific article; zbMATH DE number 1214114