Self-dual modules in characteristic two and normal subgroups

TL;DR

This paper establishes Clifford theoretic results specific to characteristic two, revealing unique correspondences between self-dual Brauer characters of finite groups and their normal subgroups, with implications for real blocks.

Contribution

It introduces new Clifford theoretic results for characteristic two, detailing the unique correspondence of self-dual Brauer characters and blocks in finite groups.

Findings

Unique self-dual Brauer character correspondence with odd multiplicity

Restriction of certain self-dual characters decomposes into distinct self-dual characters

Existence of a unique real 2-block of G covering a given real 2-block of N

Abstract

We prove Clifford theoretic results on the representations of finite groups which only hold in characteristic $2$. Let $G$ be a finite group, let $N$ be a normal subgroup of $G$ and let $\varphi$ be an irreducible $2$-Brauer character of $N$ which is self-dual. We prove that there is a unique self-dual irreducible Brauer character $\theta$ of $G$ such that $\varphi$ occurs with odd multiplicity in the restriction of $\theta$ to $N$. Moreover this multiplicity is $1$. Conversely if $\theta$ is an irreducible $2$-Brauer character of $G$ which is self-dual but not of quadratic type, the restriction of $\theta$ to $N$ is a sum of distinct self-dual irreducible Brauer character of $N$, none of which have quadratic type. Let $b$ be a real $2$-block of $N$. We show that there is a unique real $2$-block of $G$ covering $b$ which is weakly regular.