Self-dual modules in characteristic two and normal subgroups
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 be an irreducible 2-Brauer character of N which is self-dual. We prove that there is a unique self-dual irreducible Brauer character θ of G such that occurs with odd multiplicity in the restriction of θ to N. Moreover this multiplicity is 1. Conversely if θ is an irreducible 2-Brauer character of G which is self-dual but not of quadratic type, the restriction of θ 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.