Young Modules and Schur algebras

TL;DR

This paper explicitly determines the submodule structures of Young modules for symmetric groups over fields of characteristic 2 for small n, and uses this to analyze Schur algebras, including Morita equivalences.

Contribution

It provides explicit submodule structures for Young modules and Schur algebra blocks for small n, including Gabriel quivers and Morita equivalences.

Findings

Submodule structures of Young modules for n ≤ 7

Submodule structures of indecomposable projectives for Schur algebras when n ≤ 5

Partial information on structures for n=6,7 including Gabriel quivers

Abstract

We compute explicitly the submodule structure of the Young modules for symmetric groups $S_n$ over fields of characteristic 2, when $n \le 7$. We use this information to compute the submodule structure of indecomposable projectives for the corresponding Schur algebras when $n \le 5$, and we give give partial information when n=6,7, including the Gabriel quiver and the structure of the Weyl modules. We resolve all Morita equivalences between blocks of these algebras.