Products of symmetries in unitary groups
type: Item
instance of: scholarly article
title: Products of symmetries in unitary groups
zbMATH Open document ID: 0878.20032
DOI: 10.1016/S0024-3795(97)80003-1
author: Florian Bünger
author: Frieder Knüppel
author: Klaus Nielsen
published in: Linear Algebra and its Applications
publication date: 1998-01-08T00:00:00Z
review text: Let \(K\) be a commutative field of characteristic distinct from 2, allowing an involutory automorphism with fixed field \(k\). Let \(V\) be a finite-dimensional vector space equipped with a Hermitian form. An involution in the unitary group \(U\) whose space of fixed vectors is a hyperplane is called a symmetry. Let \(G\) be the subgroup of \(U\) containing all elements \(\pi\) with \(\text{det }\pi\in\{1,-1\}\). The authors show, if the norm of \(K\) is surjective on \(k\), then \(G\) is generated by symmetries, and they determine for each \(\pi\) in \(G\), how many symmetries are needed to express \(\pi\). A large portion of the paper is devoted to the case where \(|k|=3\). Here the factorization results differ from those for larger fields. As a basic tool for the proofs the authors use two additional Hermitian forms \(s\) and \(d\). These were originally introduced by the reviewer [in Linear Multilinear Algebra 35, No. 1, 11-35 (1993; Zbl 0789.20049)], which deals with the same questions as the paper under review but with the restriction \(|K|<\infty\) and \(|k|>3\).
Mathematics Subject Classification ID: 20H20
Mathematics Subject Classification ID: 51F25
Mathematics Subject Classification ID: 15A23
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 11E39
zbMATH DE Number: 1040487
zbMATH Keywords: involutory automorphisms
zbMATH Keywords: involutions
zbMATH Keywords: unitary groups
zbMATH Keywords: hyperplanes
zbMATH Keywords: groups generated by symmetries
zbMATH Keywords: factorizations
zbMATH Keywords: hermitian forms
reviewed by: Erich W. Ellers
MaRDI profile type: Publication
cites work: Products of reflections in 𝑈(𝑝,𝑞)
cites work: The reflection length of a transformation in the unitary group over a finite field
cites work: Q3997012
cites work: Products of simple isometries of given conjugacy types
cites work: On the conjugacy classes in the unitary, symplectic and orthogonal groups
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description: scientific article; zbMATH DE number 1040487