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

The Laplacian of a (weighted) Cayley graph on the Weyl group $W(B_n)$ is a $N\times N$ matrix with $N=2^{n}n!$ 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 $2n \times 2n$matrixassociatedtoa$2n$-dimensionalpermutationrepresentationof$W_n$\mathrm{.}Thisresultcanbeviewedasanextensionto$W(B_n)$ of an analogous result valid for the symmetric group, known as `Aldous’ spectral gap conjecture’, proven in 2010 by Caputo, Liggett and Richthammer.