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Generalized regular representations of big wreath products

Eugene Strahov

Abstract

Let $G$ be a finite group with $k$ conjugacy classes, and $S(\infty)$ be the infinite symmetric group, i.e. the group of finite permutations of $\left\{1,2,3,\ldots\right\}$. Then the wreath product $G_{\infty}=G\sim S(\infty)$ of $G$ with $S(\infty)$ (called the big wreath product) can be defined. The group $G_{\infty}$ is a generalization of the infinite symmetric group, and it is an example of a ``big'' group, in Vershik's terminology. For such groups the two-sided regular representations are irreducible, the conventional scheme of harmonic analysis is not applicable, and the problem of harmonic analysis is a nontrivial problem with connections to different areas of mathematics and mathematical physics. Harmonic analysis on the infinite symmetric group was developed in the works by Kerov, Olshanski, and Vershik, and Borodin and Olshanski. The goal of this paper is to extend this theory to the case of $G_{\infty}$. In particular, we construct an analogue $\mathfrak{S}_{G}$ of the space of virtual permutations. We then formulate and prove a theorem characterizing all central probability measures on $\mathfrak{S}_{G}$, and introduce generalized regular representations $T_{z_1,\ldots,z_k}$ of the big wreath product $G_{\infty}$. The paper solves a natural problem of harmonic analysis for the big wreath products: our results describe the decomposition of $T_{z_1,\ldots,z_k}$ into irreducible components.

Generalized regular representations of big wreath products

Abstract

Let be a finite group with conjugacy classes, and be the infinite symmetric group, i.e. the group of finite permutations of . Then the wreath product of with (called the big wreath product) can be defined. The group is a generalization of the infinite symmetric group, and it is an example of a ``big'' group, in Vershik's terminology. For such groups the two-sided regular representations are irreducible, the conventional scheme of harmonic analysis is not applicable, and the problem of harmonic analysis is a nontrivial problem with connections to different areas of mathematics and mathematical physics. Harmonic analysis on the infinite symmetric group was developed in the works by Kerov, Olshanski, and Vershik, and Borodin and Olshanski. The goal of this paper is to extend this theory to the case of . In particular, we construct an analogue of the space of virtual permutations. We then formulate and prove a theorem characterizing all central probability measures on , and introduce generalized regular representations of the big wreath product . The paper solves a natural problem of harmonic analysis for the big wreath products: our results describe the decomposition of into irreducible components.
Paper Structure (52 sections, 21 theorems, 225 equations, 5 figures)

This paper contains 52 sections, 21 theorems, 225 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and formulation of the problem
  3. Summary of results
  4. The space of $G$-virtual permutations $\mathfrak{S}_{G}$
  5. Central measures
  6. The generalized regular representation of $G_{\infty}=G\sim S(\infty)$
  7. The formula for the character $\chi_{z_1,\ldots,z_k}$ of $T_{z_1,\ldots,z_k}$
  8. The spectral measures
  9. Correlation functions
  10. Remarks on related works
  11. The space $\mathfrak{S}_{G}$ of G-virtual permutations
  12. The wreath product $G_n=G\sim S(n)$
  13. The canonical projection
  14. Definition of $\mathfrak{S}_{G}$
  15. Central measures on $\mathfrak{S}_{G}$
  16. ...and 37 more sections

Key Result

Proposition 2.1

The projection $p_{n,n+1}: G\sim S(n+1)\longrightarrow G\sim S(n)$ is equivariant with respect to the two-sided action of $G\sim S(n)$, i.e. for each $(g,s)\in G\sim S(n+1)$, and each $(\kappa,\pi)\in G\sim S(n)$, $(h,t)\in G\sim S(n)$.

Figures (5)

  • Figure 1: An element $\left(\left(g_1,\ldots,g_n\right),s\right)$ as a bipartite graph. The symbols $g_1$, $\ldots$, $g_n$ can be understood as the weights of the corresponding edges.
  • Figure 2: The multiplication of two group elements in terms of bipartite graphs.
  • Figure 3: The definition of the canonical projection $p_{n,n+1}$. In this example $n=6$, and the original element of $G\sim S(7)$ is $\left(\left(g_1,g_2,g_3,g_4,g_5,g_6,g_7\right),(13)(26475)\right)$. The cycle including $n+1=7$ is $2\rightarrow 6\rightarrow 4\rightarrow 7\rightarrow 5$, and $g_{n+1}=g_7$, $g_{i_m}=g_4$, $g_{i_{m+1}}=g_5$. We add the extra edge (the red dashed line) connecting the vertices $7$ and $g_7$. As a result we obtain a graph with an edge connecting $g_5$ with $4$, and passing through $g_7$. The weight of this edge is $g_5g_7$. Thus we have $p_{6,7}\left(\left(\left(g_1,g_2,g_3,g_4,g_5,g_6,g_7\right),(13)(26475)\right)\right) =\left(\left(g_1,g_2,g_3,g_4,g_5g_7,g_6\right),(13)(2645)\right).$
  • Figure 4: The maps between the representation spaces
  • Figure 5: The equivalence of inductive limits

Theorems & Definitions (48)

  • Proposition 2.1
  • proof
  • Definition 3.1
  • Theorem 3.2
  • Definition 3.3
  • Theorem 4.1
  • proof
  • Proposition 4.2
  • proof
  • Proposition 4.3
  • ...and 38 more