# A computational approach to the Frobenius-Schur indicators of finite exceptional groups

## TL;DR

This paper determines the Frobenius-Schur indicators of irreducible characters in certain finite exceptional groups, showing most are real-valued and identifying specific non-real characters, including a unique case in Ree groups.

## Contribution

It provides a complete classification of Frobenius-Schur indicators for irreducible characters of several finite exceptional groups and Ree groups, including explicit listings.

## Key findings

- No irreducible characters of F_4(q), E_7(q)_{ad}, E_8(q) have indicator -1.
- Exact list of non-real-valued characters in these groups.
- Only one character in Ree groups has indicator -1, the cuspidal unipotent character χ_{21}.

## Abstract

We prove that the finite exceptional groups $F_4(q)$, $E_7(q)_{\mathrm{ad}}$, and $E_8(q)$ have no irreducible complex characters with Frobenius-Schur indicator $-1$, and we list exactly which irreducible characters of these groups are not real-valued. We also give an exact list of complex irreducible characters of the Ree groups ${^2 F_4}(q^2)$ which are not real-valued, and we show the only character of this group which has Frobenius-Schur indicator $-1$ is the cuspidal unipotent character $\chi_{21}$ found by M. Geck.