Methods to decide if simple self-dual modules over fields of characteristic \(2\) are of quadratic type
type: Item
instance of: scholarly article
title: Methods to decide if simple self-dual modules over fields of characteristic \(2\) are of quadratic type
zbMATH Open document ID: 0839.20008
author: Rod Gow
author: Wolfgang Willems
published in: Journal of Algebra
publication date: 1996-04-08T00:00:00Z
review text: Let \(G\) be a finite group and let \(K\) be a perfect field of characteristic 2. A self-dual \(KG\)-module \(V\) admits a non degenerate \(G\)-invariant bilinear form \(B\) on \(V\times V\). By a lemma of Fong, if \(V\) is in addition simple, then \(B\) may be chosen to be alternating. If \(B\) is the polarization of a quadratic form, then \(V\) is called of quadratic type. The authors discuss two methods to decide if a simple self-dual module \(V\) is of quadratic type. The first method gives that if there is an extension \(W\) of the simple self-dual module \(V\) by the trivial module such that \(W\) is of quadratic type, then, under an additional technical hypothesis, \(V\) is not of quadratic type. The second method uses integral representations to give a positive criterion for a simple self-dual \(KG\)-module \(V\) to be of quadratic type. The authors give many applications to adjoint representations of Chevalley groups.
reviewed by: Alexander Zimmermann
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 20C10
Mathematics Subject Classification ID: 11E04
Mathematics Subject Classification ID: 20C33
zbMATH DE Number: 794355
zbMATH Keywords: finite groups
zbMATH Keywords: perfect fields
zbMATH Keywords: self-dual \(KG\)-modules
zbMATH Keywords: quadratic forms
zbMATH Keywords: simple self-dual modules
zbMATH Keywords: quadratic type
zbMATH Keywords: integral representations
zbMATH Keywords: adjoint representations of Chevalley groups
MaRDI profile type: Publication
full work available at URL: jabr.1995.1227
OpenAlex ID: W2059096038
DOI: 10.1006/JABR.1995.1227
description: scientific article; zbMATH DE number 794355