Real Classes of Finite Special Unitary Groups

TL;DR

This paper classifies real and strongly real conjugacy classes in finite special unitary groups, relating them to similar classes in special linear and orthogonal groups, with specific classifications depending on congruence conditions.

Contribution

It provides a comprehensive classification of real and strongly real classes in $SU_n(q)$ and relates these to classes in orthogonal groups, extending previous classifications.

Findings

Classification of real classes in $SU_n(q)$ for most cases.

Relation between strong reality in $SU_n(q)$ and $SO^ u_n(q)$.

Explicit classification for $SO^ u_n(q)$ when $q$ is odd and $n mod 4=2$.

Abstract

We classify all real and strongly real classes of the finite special unitary group $SU_n(q)$. Unless $q \equiv 3 (mod 4)$ and $n |4$, the classification of real classes is similar to that of the finite special linear group $SL_n(q)$. We relate strong reality in $SU_n(q)$ to strong reality in the finite special orthogonal groups $SO^\pm_n(q)$, and we classify real and strongly real classes in the last case for the group $SO^\pm_n(q)$, when q is odd and $n \equiv 2 (mod 4)$.