Product of two involutions in quaternionic special linear group

TL;DR

This work addresses reversibility questions in the quaternionic matrix groups $SL(n,\mathbb{H})$ and $PSL(n,\mathbb{H})$. It leverages Weyr canonical form, complex embeddings of quaternionic matrices, and centralizer/reversing-symmetry techniques to classify reversible and strongly reversible elements in $SL(n,\mathbb{H})$, obtaining a criterion based on even multiplicities of certain unit-modulus non-real eigenvalue blocks. The authors prove that every reversible element of $SL(n,\mathbb{H})$ is a product of two skew-involutions, and they derive the corresponding PSL classification showing reversibility corresponds to a product of two involutions in $PSL(n,\mathbb{H})$; they also connect these results to the structure of reversing symmetries and provide explicit constructions for key Jordan forms. These results complete a broad reversibility taxonomy for quaternionic linear groups and have implications for the associated dynamics and geometry of quaternionic projective spaces.

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).

Theorems & Definitions (33)

• Theorem 1.1
• Theorem 1.2: GLM1
• Theorem 1.3
• Theorem 1.4
• Definition 2.1
• Definition 2.2: rodman
• Definition 2.3: rodman
• Lemma 2.4: rodman
• Definition 2.5: COV
• Definition 2.6: GM