Products of two involutions in orthogonal and symplectic groups

TL;DR

This paper characterizes bireflectional elements in symplectic groups over fields of characteristic not 2 using Wall invariants, and provides simplified proofs for related results in orthogonal and general linear groups.

Contribution

It offers a new characterization of bireflectional elements in symplectic groups and simplifies existing proofs for orthogonal and general linear groups.

Findings

Characterization of bireflectional elements in symplectic groups using Wall invariants.

Simplified proof of Wonenburger's result for orthogonal and general linear groups.

Over fields of characteristic 2, all symplectic group elements are bireflectional.

Abstract

An element of a group is called bireflectional when it is the product of two involutions of the group (i.e. elements of order 1 or 2). If an element is bireflectional then it is conjugated to its inverse. It is known that all elements of orthogonal groups of quadratic forms are bireflectional. F. B\"unger has characterized the elements of unitary groups (over fields of characteristic not $2$) that are bireflectional. Yet in symplectic groups over fields with characteristic different from 2, in general there are elements that are conjugated to their inverse but are not bireflectional (however, over fields of characteristic 2, every element of a symplectic group is bireflectional). In this article, we characterize the bireflectional elements of symplectic groups in terms of Wall invariants, over fields of characteristic not 2: the result is cited without proof in B\"unger's PhD thesis, and attributed to Klaus Nielsen. We also take advantage of our approach to give a simplified proof of Wonenburger's corresponding result for orthogonal groups and general linear groups.