Products of symmetries in unitary groups
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 \(|k|>3\).
scientific article; zbMATH DE number 1040487
scientific article