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

TL;DR

This paper extends the Aldous spectral gap conjecture to the Weyl group $W(B_n)$, showing that the spectral gap of certain Cayley graphs can be computed from a smaller permutation representation matrix.

Contribution

It proves that for specific generating sets, the spectral gap of Cayley graphs on $W(B_n)$ equals that of a smaller permutation representation, generalizing a known result for symmetric groups.

Findings

Spectral gap equals that of a $2n imes 2n$ matrix for certain generating sets.

Extension of Aldous' spectral gap conjecture to Weyl groups $W(B_n)$.

Provides a method to compute spectral gaps more efficiently for these groups.

Abstract

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 $2n \times 2n$ matrix associated to a $2n$-dimensional permutation representation of $W_n$. This result can be viewed as an extension to $W(B_n)$ of an analogous result valid for the symmetric group, known as `Aldous' spectral gap conjecture', proven in 2010 by Caputo, Liggett and Richthammer.