{{https://www.wm.edu/as/mathematics/faculty-directory/trefethen_s.php, Stephen Trefethen}} (William & Mary) Frobenius--Schur indicators of finite exceptional groups.
Full Description
Abstract: Let G be a finite group. The Frobenius--Schur indicator of an irreducible character $\chi$, denoted $\varepsilon(\chi)$, is defined as $\varepsilon(\chi) = \frac{1}{|G|} \sum_{g \in G} \chi(g^2)$. It is known that $\varepsilon(\chi)=1, -1,$ or $0$, where $\varepsilon(\chi) = 0$ precisely when $\chi$ is not real-valued. When $\chi$ is real-valued, $\varepsilon(\chi) = 1$ if $\chi$ is afforded by a representation that may be defined over the real numbers, otherwise $\varepsilon(\chi) = -1$. In this talk we outline a computational method used to prove that the exceptional groups $F_4(q)$, $E_7(q)_{\textrm{ad}}$, and $E_8(q)$ have no irreducible characters with Frobenius--Schur indicator -1.
