Endomorphism rings of some Young modules

Abstract

Let ${\Sigma_r}$ be the symmetric group acting on ${r}$ letters, ${K}$ be a field of characteristic 2, and ${\lambda}$ and ${\mu}$ be partitions of ${r}$ in at most two parts. Denote the permutation module corresponding to the Young subgroup ${\Sigma_\lambda}$, in ${\Sigma_r}$, by ${M^\lambda}$, and the indecomposable Young module by ${Y^\mu}$. We give an explicit presentation of the endomorphism algebra ${{\rm End}_{k[\Sigma_r]}(Y^\mu)}$ using the idempotents found by Doty et al. (J Algebra 307(1):377–396, 2007).