Endomorphism rings of permutation modules over maximal Young subgroups

Abstract

Let $K$ be a field of characteristic two, and let $\lambda$ be a two-part partition of some natural number $r$. Denote the permutation module corresponding to the (maximal) Young subgroup $\Sigma_\lambda$ in $\Sigma_r$ by $M^\lambda$. We construct a full set of orthogonal primitive idempotents of the centraliser subalgebra $S_K(\lambda) = 1_\lambda S_K(2,r) 1_\lambda = End_{K\Sigma_r}(M^\lambda)$ of the Schur algebra $S_K(2,r)$. These idempotents are naturally in one-to-one correspondence with the 2-Kostka numbers.