Spectral gap for the signed interchange process with arbitrary sets

In 2020, F. Cesi introduced a random walk on the hyperoctahedral group $B_n$ and analysed its spectral gap when the allowed generators are transpositions and diagonal elements corresponding to singletons. In this paper we extend the allowed generators to transpositions and any diagonal elements, and characterise completely the set of representations from which the spectral gap arises. This settles a conjecture posed in Cesi’s paper.