On the quadratic type of some simple self-dual modules over fields of characteristic two
type: Item
instance of: scholarly article
title: On the quadratic type of some simple self-dual modules over fields of characteristic two
zbMATH Open document ID: 0888.20004
published in: Journal of Algebra
publication date: 1998-06-07T00:00:00Z
review text: Let \(K\) be an algebraically closed field of characteristic 2, \(G\) a finite group and \(V\) a nontrivial simple self-dual \(KG\)-module. Then \(V\) carries a nondegenerate \(G\)-invariant alternating bilinear form \(F\). The authors consider the question whether \(F\) is the polarization of a nondegenerate \(G\)-invariant quadratic form on \(V\). In this case they say that \(V\) is of quadratic type. They show that a finite Chevalley group defined over a finite field \(\mathbb{F}_q\) with \(q\equiv 1\pmod 4\) always has a nontrivial simple self-dual module which is not of quadratic type and which occurs with multiplicity 1 in the reduction mod 2 of the Steinberg module. They also investigate self-dual composition factors of permutation modules for the groups \(\text{PSL}_n(q)\), \(\text{PSU}_3(q)\), \(\text{GL}_3(q)\) and \(\text{GL}_4(q)\) and for odd prime powers \(q\) in more detail.
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 20C20
zbMATH DE Number: 1088620
zbMATH Keywords: finite groups
zbMATH Keywords: alternating bilinear forms
zbMATH Keywords: quadratic forms
zbMATH Keywords: finite Chevalley groups
zbMATH Keywords: simple self-dual modules
zbMATH Keywords: Steinberg modules
zbMATH Keywords: self-dual composition factors
zbMATH Keywords: permutation modules
author: Rod Gow
author: Wolfgang Willems
reviewed by: Burkhard Külshammer
MaRDI profile type: Publication
full work available at URL: jabr.1997.7048
OpenAlex ID: W2051596963
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DOI: 10.1006/JABR.1997.7048
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description: scientific article; zbMATH DE number 1088620