Commutators of symmetries in characteristic 2
type: Item
instance of: scholarly article
title: Commutators of symmetries in characteristic 2
zbMATH Open document ID: 1210.15012
author: Erich W. Ellers
author: Oliver Villa
published in: Linear Algebra and its Applications
publication date: 2011-03-04T00:00:00Z
review text: Let \(V\) be a vector space of dimension \(n\) over a field \(K\) where \(|K|>2\) equipped with a quadratic form \(Q\). Then \(Q(v+w) = Q(v)+Q(w) + f(v,w)\) for some bilinear form \(f\). The space \(V\) is called nonsingular if \(V\cap V^\perp = 0\) where the orthogonal complement \(\perp\) is with respect to \(f\). The main result of the paper asserts the following length theorem: Let \(V\) be a nonsingular vector space over a field \(K\) of characteristic \(2\) with \(|K|>3\). Suppose \(K\) is perfect and \(\pi\in SO(V)\) (the special orthogonal group) with \(\dim B(\pi) =2d\), where the path of \(\pi\) is \(B(\pi): =\text{im} (\pi-1)\). Then the length of \(\pi\) with respect to the symmetry commutators is \(d\) if \(B(\pi)\) is not totally isotropic; otherwise it is \(d+1\). The proof requires an analysis different from the case of characteristic \(2\).
Mathematics Subject Classification ID: 15A23
Mathematics Subject Classification ID: 15A63
Mathematics Subject Classification ID: 15A03
zbMATH DE Number: 5860486
zbMATH Keywords: factorization
zbMATH Keywords: commutator
zbMATH Keywords: quadratic form
zbMATH Keywords: singular vector
zbMATH Keywords: vector space
reviewed by: Tin-Yau Tam
MaRDI profile type: Publication
full work available at URL: j.laa.2010.11.031
OpenAlex ID: W1995576774
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DOI: 10.1016/J.LAA.2010.11.031
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description: scientific article; zbMATH DE number 5860486