Products of involutions in symplectic groups over general fields (I)

Abstract

Let $s$ be an $n$-dimensional symplectic form over an arbitrary field with characteristic not $2$, with $n>2$. The simplicity of the group $\mathrm{Sp}(s)/\{\pm \mathrm{id}\}$ and the existence of a non-trivial involution in $\mathrm{Sp}(s)$ yield that every element of $\mathrm{Sp}(s)$ is a product of involutions. Extending and improving recent results of Awa, de La Cruz, Ellers and Villa with the help of a completely new method, we prove that if the underlying field is infinite, every element of $\mathrm{Sp}(s)$ is the product of four involutions if $n$ is a multiple of $4$, and of five involutions otherwise. The first part of this result is shown to be optimal for all multiples of $4$ and all fields, and is shown to fail for the fields with three elements and for $n=4$. Whether the second part of the result is optimal remains an open question. Finite fields will be tackled in a subsequent article.