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

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.

Version 2 has some corrections to Table A.1 for the case q=2, added exceptional cases to Lemmas 5.1 and 5.2, and updated Section 7