Title: Spectral Gaps and Symmetric Groups: From Aldous' Conjecture to Signed Permutations

Abstract:
Aldous' spectral gap conjecture, proven by Caputo, Liggett, and Richthammer in 2010, reveals a striking connection between the spectral gap of certain Cayley graphs on the symmetric group S_n and the representation theory of the group. Specifically, for Cayley graphs generated by transpositions, the spectral gap coincides with the first nontrivial eigenvalue of the Laplacian associated to a significantly smaller graph arising from the standard representation of S_n. In this talk, I will outline the key ideas behind this result and the role played by representation theory in reducing the complexity of spectral gap computations. I will then discuss a parallel result due to Cesi, who extended this framework to the group of signed permutations, offering analogous insights into Cayley graphs on W_n.