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

2022/11/18 by Deng, Yongtao, Tan, Shi Jie Samuel
#20B05 #20B25 #60J10 #FOS: Mathematics #Group Theory (math.GR) #Probability (math.PR) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2211.10462

Abstract

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 ≤ 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 ≥ 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.

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