Some Reality Properties of Finite Simple Orthogonal Groups

TL;DR

This paper investigates the reality properties of finite simple orthogonal groups, establishing conditions under which conjugacy classes are strongly or weakly real, and analyzing the Frobenius-Schur indicators of their characters.

Contribution

It proves that all real conjugacy classes are strongly real in most cases and constructs weakly real classes in specific exceptional cases, also analyzing character indicators.

Findings

All real classes are strongly real in most cases.

Weakly real classes exist in specific exceptional cases.

No irreducible character has Frobenius-Schur indicator -1, except possibly in an exceptional case.

Abstract

We prove several reality properties for finite simple orthogonal groups. For any prime power $q$ and $m\geq 1$, we show that all real conjugacy classes are strongly real in the simple groups $\mathrm{P}\Omega^{\pm}(4m+2,q), m \geq 1$, except in the case $\mathrm{P}\Omega^{-}(4m+2,q)$ with $q \equiv 3(\mathrm{mod} \; 4)$, and we construct weakly real classes in this exceptional case for any $m$. We also show that no irreducible complex character of $\mathrm{P}\Omega^{\pm}(n,q)$ can have Frobenius-Schur indicator $-1$, except possibly in the case $\mathrm{P}\Omega^{-}(4m+2,q)$ with $q \equiv 3(\mathrm{mod} \; 4)$.