Products of skew-involutions

Abstract

It is shown that every $2n$-by-$2n$ matrix over a field $\mathbb{F}$ with determinant 1 is a product of (i) four or fewer skew-involutions ($A^2 = -I$) provided $\mathbb{F} \neq \mathbb{Z}_3$, and (ii) eight or fewer skew-involutions if $\mathbb{F} = \mathbb{Z}_3$ and $n > 1$. Every real symplectic matrix is a product of six real symplectic skew-involutions, and an explicit factorization of a complex symplectic matrix into two symplectic skew-involutions is given.