Aldous' spectral gap conjecture for normal sets

Aldous' spectral gap conjecture, proved in 2009 by Caputo, Liggett and Richthammer, states the following a priori very surprising fact: the spectral gap of a random walk on a finite graph is equal to the spectral gap of the interchange process on the same graph.

This seminal result has a very natural interpretation in Representation Theory, which leads to natural conjectural generalizations. In joint works with Gadi Kozma and Ori Parzanchevski we study some of these possible generalizations, clarify the picture in the case of normal generating sets and reach a more refined, albeit bold, conjecture.