Hooks generate the representation ring of the symmetric group

TL;DR

This paper proves that the representation ring of the symmetric group is generated by exterior powers of its natural representation, using a simple formula by Y. Dvir, and explores potential generalizations to other reflection groups.

Contribution

It introduces a new generating set for the symmetric group's representation ring and demonstrates a simple proof leveraging Dvir's formula, with potential extensions to other reflection groups.

Findings

Representation ring generated by exterior powers

Simple proof using Dvir's formula

Potential generalization to other reflection groups

Abstract

We prove that the representation ring of the symmetric group on $n$ letters is generated by the exterior powers of its natural $(n-1)$-dimensional representation. The proof we give illustrates a strikingly simple formula due to Y. Dvir. We provide an application and investigate a possible generalization of this result to some other reflection groups.