Endomorphism Rings of Some Young Modules

TL;DR

This paper explicitly describes the endomorphism algebra of certain Young modules over symmetric groups in characteristic 2, using specific idempotents, advancing understanding of their algebraic structure.

Contribution

It provides an explicit presentation of the endomorphism algebra of Young modules for symmetric groups in characteristic 2, utilizing idempotents from prior research.

Findings

Explicit presentation of Endomorphism algebra of Young modules

Use of idempotents from Doty, Erdmann, and Henke

Enhanced understanding of module structure in characteristic 2

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, Erdmann and Henke in [1].