Strongly real classes in finite unitary groups of odd characteristic

TL;DR

This paper classifies strongly real conjugacy classes in finite unitary groups of odd characteristic, linking them to orthogonal groups and providing formulas and partial results for related groups.

Contribution

It provides a complete classification of strongly real classes in finite unitary groups over odd fields and connects these classes to orthogonal groups, with additional partial results for symplectic groups.

Findings

Strongly real classes correspond to elements in embedded orthogonal groups.

A characterization involving elementary divisors of the form (t ± 1)^{2m}.

Derived a generating function for counting strongly real classes.

Abstract

We classify all strongly real conjugacy classes of the finite unitary group $\U(n, F_q)$ when $q$ is odd. In particular, we show that $g \in \U(n, F_q)$ is strongly real if and only if $g$ is an element of some embedded orthogonal group $O^{\pm}(n, F_q)$. Equivalently, $g$ is strongly real in $\U(n, F_q)$ if and only if $g$ is real and every elementary divisor of $g$ of the form $(t \pm 1)^{2m}$ has even multiplicity. We apply this to obtain partial results on strongly real classes in the finite symplectic group $\Sp(2n, F_q)$, $q$ odd, and a generating function for the number of strongly real classes in $\U(n, F_q)$, $q$ odd, and we also give partial results on strongly real classes in $\U(n, F_q)$ when $q$ is even.