Bireflectionality of the orthogonal group over a valuation domain
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.
scientific article; zbMATH DE number 63614