The quadratic type of certain irreducible modules for the symmetric group in characteristic two. (Q1882876)

In characteristic 2 an irreducible module \(D\) for a finite group \(G\) always carries a \(G\)-invariant non-degenerate symplectic form by Fong's lemma, i.e., \(D\) is of symplectic type. However, some modules may even have a \(G\)-invariant non-degenerate quadratic form, i.e., \(D\) is of quadratic type. To check this is usually a nontrivial task and affords often tricky arguments. Let \(D^\lambda\) be the irreducible module over \(\text{GF}(2)\) of the symmetric group \(S_n\) corresponding to a 2-regular partition \(\lambda\). In the paper under review the authors prove the following two results. Theorem 1: Let \(r\) be a non-negative integer and let \(\lambda=(n-2^r,2^r)\), where \(2^{r+1}<n\). Then \(D^\lambda\) is not of quadratic type if and only if \(n\equiv k\bmod{r^{r+2}}\), where \(k\in\{2^{r+1}+2^r-1,\dots,2^{r-2}-2\}\). Theorem 2: Let \(\lambda=(n,n-m)\) be a 2-regular partition, where \(m\geq 3\) is an integer which is not a power of 2. Then \(D^\lambda\) is of quadratic type. -- The proofs are mainly based on lattice considerations and a formula of Schaper [Diplom\-ar\-beit, Bonn (1981)]. In the final end the question of the existence of an \(S_n\)-invariant non-degenerate quadratic form turns out to be merely a question in combinatorics.