On the quadratic type of some simple self-dual modules over fields of characteristic two
Let \(K\) be an algebraically closed field of characteristic 2, \(G\) a finite group and \(V\) a nontrivial simple self-dual \(KG\)-module. Then \(V\) carries a nondegenerate \(G\)-invariant alternating bilinear form \(F\). The authors consider the question whether \(F\) is the polarization of a nondegenerate \(G\)-invariant quadratic form on \(V\). In this case they say that \(V\) is of quadratic type. They show that a finite Chevalley group defined over a finite field \(\mathbb{F}_q\) with \(q\equiv 1\pmod 4\) always has a nontrivial simple self-dual module which is not of quadratic type and which occurs with multiplicity 1 in the reduction mod 2 of the Steinberg module. They also investigate self-dual composition factors of permutation modules for the groups \(\text{PSL}_n(q)\), \(\text{PSU}_3(q)\), \(\text{GL}_3(q)\) and \(\text{GL}_4(q)\) and for odd prime powers \(q\) in more detail.
scientific article; zbMATH DE number 1088620
scientific article