On products of two involutions in the orthogonal group of a vector space
type: Item
instance of: scholarly article
title: On products of two involutions in the orthogonal group of a vector space
zbMATH Open document ID: 0623.20035
DOI: 10.1016/0024-3795(87)90091-7
author: Frieder Knüppel
author: Klaus Nielsen
published in: Linear Algebra and its Applications
publication date: 1987-01-01T00:00:00Z
review text: Let V be a vector space over a commutative field K such that the characteristic of K is distinct from 2. Assume that V is endowed with a symmetric nondegenerate bilinear form. For each \(\pi\) in the orthogonal group O(V) the subspace \(B(\pi)=image(\pi -1)\) is called the path of \(\pi\). The authors show: (A) If \(\rho\) and \(\sigma\) are involutions in O(V) and \(\pi =\rho \sigma\), then there is an orthogonal decomposition of V into orthogonally indecomposable \(\pi\)-modules that are invariant under \(\rho\) and \(\sigma\) ; (B) If \(\pi\in O(V)\), then there are involutions \(\rho\),\(\sigma\in O(V)\) such that \(\pi =\rho \sigma\) and \(B(\pi)=B(\rho)+B(\sigma)\) holds if and only if an orthogonal decomposition of V into orthogonally indecomposable \(\pi\)-modules does not contain a term whose minimal polynomial is \((x-1)^{\alpha}\) where \(\alpha\) is even.
Mathematics Subject Classification ID: 20H20
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 15A23
Mathematics Subject Classification ID: 51F25
zbMATH DE Number: 4010778
zbMATH Keywords: orthogonal group
zbMATH Keywords: path
zbMATH Keywords: involutions
zbMATH Keywords: orthogonal decomposition
zbMATH Keywords: orthogonally indecomposable \(\pi \)-modules
reviewed by: Erich W. Ellers
MaRDI profile type: Publication
full work available at URL: 0024-3795(87)90091-7
OpenAlex ID: W2013518422
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description: scientific article; zbMATH DE number 4010778