$k$-involutions of $\text{SL}(n,k)$ over Fields of Characteristic 2

TL;DR

This paper classifies and provides explicit matrix representatives for all $k$-involutions of $ ext{SL}(n,k)$ over fields of characteristic 2, aiding the understanding of symmetric $k$-varieties in this setting.

Contribution

It offers a complete classification and explicit matrix representatives of $k$-involutions of $ ext{SL}(n,k)$ over fields of characteristic 2, including fixed point groups.

Findings

Explicit matrix representatives for each isomorphy class of $k$-involutions.

Description of fixed point groups for each involution type.

Complete classification over fields of characteristic 2.

Abstract

Symmetric $k$-varieties generalize Riemannian sym\-me\-tric spaces to reductive groups defined over arbitrary fields. For most perfect fields, it is known that symmetric $k$-varieties are in one-to-one correspondence with isomorphy classes of $k$-involutions. Therefore, it is useful to have representatives of each isomorphy class in order to describe the $k$-varieties. Here we give matrix representatives for each isomorphy class of $k$-involutions of $\text{SL}(n,k)$ in the case that $k$ is any field of characteristic 2; we also describe fixed point groups of each type of involution.