On involutions and indicators of finite orthogonal groups

Abstract

We study the numbers of involutions and their relation to Frobenius-Schur indicators in the groups $\mathrm{SO}^{\pm}(n,q)$ and $\Omega^{\pm}(n,q)$. Our point of view for this study comes from two motivations. The first is the conjecture that a finite simple group $G$ is strongly real (all elements are conjugate to their inverses by an involution) if and only if it is totally orthogonal (all Frobenius-Schur indicators are 1), and we are able to show this holds for all finite simple groups $G$ other than the groups $\mathrm{Sp}(2n,q)$ with $q$ even or $\Omega^{\pm}(4m,q)$ with $q$ even. We prove computationally that for small $n$ and $m$ this statement indeed holds for these groups by equating their character degree sums to the number of involutions. We also prove a result on a certain twisted indicator for the groups $\mathrm{SO}^{\pm}(4m+2,q)$ with $q$ odd. Our second motivation is to continue the work of Fulman, Guralnick, and Stanton on generating function and asymptotics for involutions in classical groups. We extend their work by finding generating functions for the numbers of involutions in $\mathrm{SO}^{\pm}(n,q)$ and $\Omega^{\pm}(n,q)$ for all $q$, and we use these to compute the asymptotic behavior for the number of involutions in these groups when $q$ is fixed and $n$ grows.