Ordering the Representations of ${{S}_{n}}$ Using the Interchange Process

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 $\prec $ 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 for this order.