Nikolaus conference 2016

Let \(G\) be a simple linear algebraic group over an algebraically closed field \(K\) of characteristic \(p \geq 0\) and let \(V\) be an irreducible rational \(G\)-module with highest weight \(\lambda\). When \(V\) is self-dual, a basic question to ask is whether \(V\) has a non-degenerate \(G\)-invariant alternate bilinear form or a non-degenerate \(G\)-invariant quadratic form. If \(p \neq 2\), the answer is well known and easily described in terms of \(\lambda\). In the case where \(p = 2\), we know that if \(V\) is self-dual, it always has a non-degenerate \(G\)-invariant alternate bilinear form. However, determining when \(V\) has a non-degenerate \(G\)-invariant quadratic form is a classical problem that still remains open. I will present some recent results which settle the problem for some families of \(\lambda\). For example, an answer can be given in the case where \(G\) is of classical type and \(\lambda\) is a fundamental highest weight.