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Probability measures on families of partitions related to harmonic analysis on big wreath products

Eugene Strahov

TL;DR

The paper develops a harmonic-analysis framework for the big wreath product $S_{\infty}(G)$ with a compact group $G$, constructing generalized regular representations $T_z$ and describing their characters via $z$-measures on families of partitions. It defines Ewens-type measures on wreath products $S_n(G)$, builds the space of $G$-virtual permutations $\mathfrak{S}_G$, and derives an explicit formula for the $z$-measures $M_z^{(n)}(\Lambda)$ in terms of $I=\langle z,z\rangle_G$, $\alpha(\zeta)=\langle z,\chi^{\pi^\zeta}\rangle_G$, and box statistics $c(\Box),h(\Box)$. The authors connect these measures to harmonic functions on the branching graph $\mathbb{Y}(\hat{G})$ and provide an integral (spectral) representation of the characters through generalized Thoma data, culminating in a description of the spectral $z$-measures $P_z$ and their random-parameter realization. This generalizes the Kerov–Olshanski–Vershik framework from $S_{\infty}$ to wreath products with compact groups, unifying finite and infinite $G$ cases and recovering known results while delivering new spectral-measure descriptions.

Abstract

We construct generalized regular representations of the wreath product of a compact group with the infinite symmetric group. The characters of these representations are determined by probability measures on families of partitions called the $z$-measures for the wreath product of a compact group with the symmetric group in the present paper. Our main result is an explicit formula for these $z$-measures which holds true for an arbitrary compact group. The result enables us to describe the spectral measures of the generalized regular representations of big wreath products.

Probability measures on families of partitions related to harmonic analysis on big wreath products

TL;DR

The paper develops a harmonic-analysis framework for the big wreath product with a compact group , constructing generalized regular representations and describing their characters via -measures on families of partitions. It defines Ewens-type measures on wreath products , builds the space of -virtual permutations , and derives an explicit formula for the -measures in terms of , , and box statistics . The authors connect these measures to harmonic functions on the branching graph and provide an integral (spectral) representation of the characters through generalized Thoma data, culminating in a description of the spectral -measures and their random-parameter realization. This generalizes the Kerov–Olshanski–Vershik framework from to wreath products with compact groups, unifying finite and infinite cases and recovering known results while delivering new spectral-measure descriptions.

Abstract

We construct generalized regular representations of the wreath product of a compact group with the infinite symmetric group. The characters of these representations are determined by probability measures on families of partitions called the -measures for the wreath product of a compact group with the symmetric group in the present paper. Our main result is an explicit formula for these -measures which holds true for an arbitrary compact group. The result enables us to describe the spectral measures of the generalized regular representations of big wreath products.
Paper Structure (35 sections, 17 theorems, 206 equations)

This paper contains 35 sections, 17 theorems, 206 equations.

Table of Contents

  1. Introduction
  2. Formulation of the problem and description of main results
  3. The Ewens measure on a wreath product of a compact group with the symmetric group
  4. The generalized regular representations $T_z$ of $S_{\infty}(G)$.
  5. A formula for the characters of $T_z$. The $z$-measures for the wreath products $S_{n}(G)$
  6. Description of the spectral $z$-measures
  7. Organization of the paper
  8. Notation
  9. The symmetric group and Young diagrams
  10. The wreath product $S_n\left(G\right)$.
  11. Conjugacy classes and irreducible representations of $G$
  12. Representations of $D_n(G)=G^n$
  13. The conjugacy classes of $S_n\left(G\right)$
  14. Irreducible unitary representations of $S_n\left(G\right)$
  15. The Ewens measure on $S_n\left(G\right)$
  16. ...and 20 more sections

Key Result

Theorem 1.2

Assume that an irreducible representation of $S_n(G)$ is parameterized by a collection $\Lambda$ of Young diagrams Then we have where $I=\left<z,z\right>_G$, $(I)_n=I(I+1)\ldots (I+n-1)$. The function $\alpha : \widehat{G}\longrightarrow\mathop{\mathrm{\mathbb{C}}}\nolimits$ is defined by where $\chi^{\pi^{\zeta}}$ denotes the character of the irreducible representation $\pi^{\zeta}$ of $G$ par

Theorems & Definitions (40)

  • Definition 1.1
  • Theorem 1.2
  • Example 2.1
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Remark 3.3
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • ...and 30 more