Product of two involutions in quaternionic special linear group

TL;DR

This paper classifies reversible and strongly reversible elements in quaternionic special linear and projective linear groups, showing they can be expressed as products of two involutions or skew-involutions, advancing understanding of their algebraic structure.

Contribution

It provides a complete classification of reversible and strongly reversible elements in quaternionic linear groups, linking them to products of involutions and skew-involutions.

Findings

Reversible elements are conjugate to their inverses.

Strongly reversible elements can be expressed as products of two involutions.

Characterization of reversibility in quaternionic special linear groups.

Abstract

An element of a group is called reversible if it is conjugate to its own inverse. Reversible elements are closely related to strongly reversible elements, which can be expressed as a product of two involutions. In this paper, we classify the reversible and strongly reversible elements in the quaternionic special linear group $\mathrm{SL}(n,\mathbb{H})$ and quaternionic projective linear group $ \mathrm{PSL}(n,\mathbb{H})$. We prove that an element of $ \mathrm{SL}(n,\mathbb{H})$ (resp. $ \mathrm{PSL}(n,\mathbb{H})$) is reversible if and only if it is a product of two skew-involutions (resp. involutions).