Totally orthogonal finite simple groups

Abstract

We prove that if $G$ is a finite simple group, then all irreducible complex representations of $G$ by be realized over the real numbers if and only if every element of $G$ may be written as a product of two involutions in $G$. This follows from our result that if $q$ is a power of $2$, then all irreducible complex representations of the orthogonal groups ${\mathrm{O}}^{\pm }(2n,{\mathbb{F}}_q)$ may be realized over the real numbers. We also obtain generating functions for the sums of degrees of several sets of unipotent characters of finite orthogonal groups, and we obtain a twisted version of our main result for a broad family of finite classical groups.

References in corpus (1)

• Action of Automorphisms on Irreducible Characters of Symplectic Groups