Random Walks on the Generalized Symmetric Group: Cutoff for the One-sided Transposition Shuffle

In this paper, we present a detailed proof for the exhibition of a cutoff for the one-sided transposition (OST) shuffle on the generalized symmetric group $G_{m,n}$. Our work shows that based on techniques for $m\le 2$ proven by Matheau-Raven, we can prove the cutoff in total variation distance and separation distance for an unbiased OST shuffle on $G_{m,n}$ for any fixed $m\ge 1$ in time $n\log (n)$. We also prove the branching rules for the simple modules of $G_{m,n}$ and lay down some of the mathematical foundation for proving the conjecture for the cutoff in total variation distance for any general biased OST shuffle on $G_{m,n}$.