Voting, the symmetric group, and representation theory

TL;DR

This paper explores voting systems through the lens of algebra and representation theory, modeling voting profiles as modules over the symmetric group and extending classical results with algebraic tools.

Contribution

It introduces an algebraic framework for voting theory using symmetric group modules, providing new insights and extensions to existing results.

Findings

Voting profiles can be modeled as elements of $Q S_n$-modules.

Representation theory tools like Schur's Lemma facilitate analysis of voting systems.

The approach offers a unified algebraic perspective on classical voting results.

Abstract

We show how voting may be viewed naturally from an algebraic perspective by viewing voting profiles as elements of certain well-studied $\mathbb{Q}S_n$-modules. By using only a handful of simple combinatorial objects (e.g., tabloids) and some basic ideas from representation theory (e.g., Schur's Lemma), this allows us to recast and extend some well-known results in the field of voting theory.