Endomorphism algebras of 2-row permutation modules over characteristic 3

Abstract

Given $r \in \mathbf{N},$ let $\lambda$ be a partition of $r$ with at most two parts. Let $\mathbf{F}$ be a field of characteristic 3. Write $M^\lambda$ for the $\mathbf{F}S_r$-permutation module corresponding to the action of the symmetric group $S_r$ on the cosets of the maximal Young subgroup $S_\lambda.$ We construct a full set of central primitive idempotents in $\text{End}_{\mathbf{F} S_r}(M^\lambda)$ in this case. We also determine the Young module corresponding to each primitive idempotent that we construct.