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

TL;DR

This paper proves that over infinite fields, every element in symplectic groups can be expressed as a product of four or five involutions depending on the dimension, extending previous results with a new method.

Contribution

It introduces a novel approach to decompose symplectic group elements into involutions, improving and extending prior work, and establishes optimal bounds for infinite fields.

Findings

Every element in symplectic groups over infinite fields is a product of four involutions if dimension is multiple of 4.

Otherwise, every element is a product of five involutions.

The result is optimal for multiples of 4 and all fields, with some exceptions for small fields.

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.