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

## TL;DR

This paper constructs a resolution of irreducible complex representations of the symmetric group by restricting representations from the general linear group, providing a categorification of recent results and minimal resolutions of simple modules.

## Contribution

It introduces a new method to resolve irreducible $S_n$-modules via restrictions from $GL_n(C)$, linking representation theory of symmetric groups and general linear groups.

## Key findings

- Constructed explicit resolutions of irreducible $S_n$-modules
- Categorifies recent combinatorial results of Assaf and Speyer
- Provides minimal resolutions of simple modules in the category of finite sets

## 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).