Ordering the representations of S_n using the interchange process

TL;DR

This paper introduces an ordering on the irreducible representations of the symmetric group S_n inspired by Aldous' conjecture, providing new insights into the spectral gap of the interchange process and extending the understanding of the Aldous order.

Contribution

It defines a new order on S_n representations related to Aldous' conjecture and identifies additional entries in this order, advancing the theoretical framework.

Findings

Established a new order on irreducible representations of S_n.

Connected the order to Aldous' conjecture and spectral gap analysis.

Identified new comparable representations within the order.

Abstract

Inspired by Aldous' conjecture for the spectral gap of the interchange process and its recent resolution by Caputo, Liggett and Richthammer, we define an associated order on the irreducible representations of S_n. Aldous' conjecture is equivalent to certain representations being comparable in this order, and hence determining the "Aldous order" completely is a generalized question. We show a few additional entries in this order.