Strongly real $2$-blocks and the Frobenius-Schur indicator

Abstract

Let $G$ be a finite group, let $k$ be an algebraically closed field of characteristic $2$ and let $\Omega:=\{g\in G\mid g^2=1_G\}$. It is shown that for a block $B$ of $kG$, the permutation module $k\Omega$ has a $B$-composition factor if and only if the Frobenius-Schur indicator of the regular character of $B$ is non-zero or equivalently if and only if $B$ is real with a strongly real defect class.