Methods to decide if simple self-dual modules over fields of characteristic \(2\) are of quadratic type
Let \(G\) be a finite group and let \(K\) be a perfect field of characteristic 2. A self-dual \(KG\)-module \(V\) admits a non degenerate \(G\)-invariant bilinear form \(B\) on \(V\times V\). By a lemma of Fong, if \(V\) is in addition simple, then \(B\) may be chosen to be alternating. If \(B\) is the polarization of a quadratic form, then \(V\) is called of quadratic type. The authors discuss two methods to decide if a simple self-dual module \(V\) is of quadratic type. The first method gives that if there is an extension \(W\) of the simple self-dual module \(V\) by the trivial module such that \(W\) is of quadratic type, then, under an additional technical hypothesis, \(V\) is not of quadratic type. The second method uses integral representations to give a positive criterion for a simple self-dual \(KG\)-module \(V\) to be of quadratic type. The authors give many applications to adjoint representations of Chevalley groups.
adjoint representations of Chevalley groups
scientific article; zbMATH DE number 794355