Resolving Irreducible $\mathbb{C}S_n$-Modules by Modules Restricted from $GL_n(\mathbb{C})$

Abstract

We construct a resolution of irreducible complex representations of the symmetric group $S_n$ by restrictions of representations of $GL_n(\mathbb{C})$ (where $S_n$ is the subgroup of permutation matrices). This categorifies a recent result of Assaf and Speyer. Our construction also gives minimal resolutions of simple $\mathcal{F}$-modules (here $\mathcal{F}$ is the category of finite sets).