On the number of real classes in the finite projective linear and unitary groups

TL;DR

This paper investigates the number of real conjugacy classes in finite projective linear and unitary groups, establishing equalities among these counts and providing a generating function for their common value.

Contribution

It refines existing results by showing equalities of real conjugacy class counts across various finite groups and extends these results to new group settings.

Findings

Number of real classes in PGL equals that in GL contained in SL.

Number of real classes in PGU equals that in U contained in SU.

Provides a generating function for the common count.

Abstract

We show that for any $n$ and $q$, the number of real conjugacy classes in $\mathrm{PGL}(n, \mathbb{F}_q)$ is equal to the number of real conjugacy classes of $\mathrm{GL}(n, \mathbb{F}_q)$ which are contained in $\mathrm{SL}(n, \mathbb{F}_q)$, refining a result of Lehrer, and extending the result of Gill and Singh that this holds when $n$ is odd or $q$ is even. Further, we show that this quantity is equal to the number of real conjugacy classes in $\mathrm{PGU}(n, \mathbb{F}_q)$, and equal to the number of real conjugacy classes of $\mathrm{U}(n, \mathbb{F}_q)$ which are contained in $\mathrm{SU}(n, \mathbb{F}_q)$, refining results of Gow and Macdonald. We also give a generating function for this common quantity.