On the spectral gap of some Cayley graphs on the Weyl group W(Bn)

The Laplacian of a (weighted) Cayley graph on the Weyl group W(Bn) is a N×N matrix with N=2nn! equal to the order of the group. We show that for a class of (weighted) generating sets, its spectral gap (lowest nontrivial eigenvalue), is actually equal to the spectral gap of a 2n×2n matrix associated to a 2n-dimensional permutation representation of Wn. This result can be viewed as an extension to W(Bn) of an analogous result valid for the symmetric group, known as “Aldous' spectral gap conjecture”, proven in 2010 by Caputo, Liggett and Richthammer.