Bireflectionality of the orthogonal group over a valuation domain
type: Item
instance of: scholarly article
title: Bireflectionality of the orthogonal group over a valuation domain
zbMATH Open document ID: 0779.20028
DOI: 10.1016/0021-8693(92)90019-I
author: Erich W. Ellers
author: Hiroyuki Ishibashi
published in: Journal of Algebra
publication date: 1992-09-27T00:00:00Z
review text: Let \(R\) be a valuation domain with maximal ideal \(A\) and \(V\) a free module of rank \(n\) over \(R\). A vector \(v \in V\) is called unimodular if \(v\not\in AV\). Assume \(V\) is endowed with a quadratic form \(Q: V\to R\) such that \(Q(v)\) is a unit in \(R\) for every unimodular \(v \in V\). Let \(O(V)\) denote the corresponding orthogonal group of \(V\). For \(\pi \in O(V)\) define \(F(\pi): = \text{ker}(\pi-1)\) and \(B(\pi) = (\pi - 1)V\). Theorem 1: Every element in \(O(V)\) is a product of at most two involutions. If \(\pi \in O(V)\) and \(d = n-\dim F(\pi)\) is even, then there are involutions \(\kappa,\rho \in O(V)\) such that \(\pi = \kappa\rho\) and \(\dim F(\kappa) = \dim F(\rho) = n-(d/2)\). For 2 a unit in \(R\) let \(O^ +(V) = \{\pi \in O(V)\mid\text{det }\pi = 1\}\) denote the special orthogonal group of \(V\). The proof of theorem 1 also yields a factorization of a special orthogonal transformation into half-turns (Theorem 2). The minimal number of factors is determined. The involutions used in the factorization of any transformation in the special orthogoal group can all be taken from one conjugacy class of orthogonal involutions (Theorem 3). The proofs of the theorem proceeds by induction and does not use the existence of certain normal forms of orthogonal transformations.
reviewed by: Hans-Jochen Bartels
Mathematics Subject Classification ID: 20H25
Mathematics Subject Classification ID: 15A23
Mathematics Subject Classification ID: 11E08
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 63614
zbMATH Keywords: product of involutions
zbMATH Keywords: valuation domain
zbMATH Keywords: free module
zbMATH Keywords: quadratic form
zbMATH Keywords: orthogonal group
zbMATH Keywords: special orthogonal group
zbMATH Keywords: special orthogonal transformation
zbMATH Keywords: number of factors
zbMATH Keywords: orthogonal involutions
MaRDI profile type: Publication
cites work: Bireflectionality in Classical Groups
cites work: Products of half-turns
cites work: Generators of an orthogonal group over a local valuation domain
cites work: On products of two involutions in the orthogonal group of a vector space
cites work: Q3800196
full work available at URL: 0021-8693(92)90019-i
OpenAlex ID: W2024695978
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description: scientific article; zbMATH DE number 63614