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 $S_K(\lambda )=1_{\lambda }{S}_K(2,r)1_{\lambda }=En{d}_{K{\Sigma }_r}(M^{\lambda })$ of the Schur algebra $S_K(2,r)$. These idempotents are naturally in one-to-one correspondence with the 2-Kostka numbers.

18 pages. To appear in J. of Algebra