Jucys--Murphy Elements for Wreath Products and Their Application to Dynamical Random Multi-Diagrams

Abstract

The equivalence classes of irreducible representations of wreath product $\mathfrak{S}_n(T) = T^n \rtimes \mathfrak{S}_n$ of finite group $T$ with respect to symmetric group $\mathfrak{S}_n$ are parametrized by $\mathbb{Y}_n(\widehat{T})$, the $\lvert \widehat{T}\rvert$-tuple Young diagrams with total size $n$. We show a formula connecting the Kerov transition measures of these Young diagrams with the Jucys--Murphy elements of $\mathfrak{S}_n(T)$. This formula is due to Biane in the case of symmetric groups. The formula enables us to investigate asymptotic property of the shapes of multi-diagrams through combinatorial analysis for the Jucys--Murphy elements. On the other hand, a Markov chain is introduced on $\mathbb{Y}_n(\widehat{T})$, canonically reflecting the branching rule for the tower of wreath product groups. We have a continuous time stochastic process on $\mathbb{Y}_n(\widehat{T})$ from this chain by replacing the discrete time by a counting process. Our project is to specify the deterministic limit shape of multi-diagrams at each macroscopic time through appropriate space-time scaling limit, and to describe evolution of related quantities characterizing the shape. Especially, we derive dynamical concentrated limit shapes in the case of abelian $T$ by using free probability tools under the assumption of approximate factorization property for initial ensembles with an additional property of a pausing time distribution.