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

Akihito Hora

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.

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

Abstract

The equivalence classes of irreducible representations of wreath product of finite group with respect to symmetric group are parametrized by , the -tuple Young diagrams with total size . We show a formula connecting the Kerov transition measures of these Young diagrams with the Jucys--Murphy elements of . 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 , canonically reflecting the branching rule for the tower of wreath product groups. We have a continuous time stochastic process on 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 by using free probability tools under the assumption of approximate factorization property for initial ensembles with an additional property of a pausing time distribution.
Paper Structure (13 sections, 10 theorems, 138 equations, 5 figures, 3 tables)

This paper contains 13 sections, 10 theorems, 138 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['th:1-1']}
  3. Proof of Theorem \ref{['th:1-2']}
  4. Preparatory argument
  5. Computation of the main terms for proof of Theorem \ref{['th:1-2']}
  6. Proof of Theorem \ref{['th:1-3']}
  7. Convergence of averaged moments of rescaled transition measures
  8. $t=0$ (initial ensemble)
  9. $t\geqq 0$ (any macroscopic time)
  10. $L^2$-convergence of moments of rescaled transition measures
  11. Variance of moments of Jucys--Murphy elements
  12. Final step of Proof of Theorem \ref{['th:1-3']}
  13. Typical examples

Key Result

Theorem 1.1

For $\lambda = (\lambda^\zeta)_{\zeta\in \widehat{T}} \in\mathbb{Y}_n(\widehat{T})$ and $k\in\mathbb{N}$, we have or equivalently (by inversion)

Figures (5)

  • Figure 1: Coordinates of a rectanglular diagram
  • Figure 2: $\pi = \{ \{ 1,6,9,10\}, \{2,3,4,5\}, \{7,8\}\} \in \mathrm{NC}((2^14^2)) \subset \mathrm{NC}(10)$
  • Figure 3: Left: stratification of Young diagrams $\overline{\mathbb{Y}}$, Right: operations \ref{['eq:3-1-4']} for $\sigma$
  • Figure 4: An element of $\mathrm{NC}(15)$ with 1st stratum: $\{4,5,6\}$; 2nd stratum: $\{3,7\}$, $\{8,9\}$, $\{12,13,14\}$; 3rd stratum: $\{1,2,10\}$, $\{11,15\}$
  • Figure 5: inner block(s)

Theorems & Definitions (10)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 4.1
  • Proposition 4.2
  • Proposition 4.3
  • Proposition 4.4
  • Proposition 4.5
  • Lemma 4.6
  • Lemma 4.7