Let G be a finite group with k conjugacy classes, and S(∞) be the infinite symmetric group, i.e., the group of finite permutations of {1, 2, 3, …}. Then the wreath product G∞ = G ∼ S (∞) of G with S (∞) (called the big wreath product) can be defined. The group G∞ 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∞. In particular, we construct an analogue <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11856_2025_2739_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="27" /> </InlineMediaObject> <EquationSource Format="TEX">${\frak{S}}_{G}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mrow> <mrow> <mi mathvariant="fraktur">S</mi> </mrow> </mrow> <mrow> <mi>G</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> of the space of virtual permutations. We then formulate and prove a theorem characterizing all central probability measures on <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11856_2025_2739_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="27" /> </InlineMediaObject> <EquationSource Format="TEX">${\frak{S}}_{G}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mrow> <mrow> <mi mathvariant="fraktur">S</mi> </mrow> </mrow> <mrow> <mi>G</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>. Next, we introduce generalized regular representations <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11856_2025_2739_Article_IEq3.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="248" /> </InlineMediaObject> <EquationSource Format="TEX">$\{T_{z_{1}},\cdots,z_{k} : z_{1}\in \mathbb{C},\cdots,z_{k}\in \mathbb{C}\}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>T</mi> <mrow> <msub> <mi>z</mi> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>z</mi> <mrow> <mi>k</mi> </mrow> </msub> <mo>:</mo> <msub> <mi>z</mi> <mrow> <mn>1</mn> </mrow> </msub> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>z</mi> <mrow> <mi>k</mi> </mrow> </msub> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mo fence="false" stretchy="false">}</mo> </math></EquationSource> </InlineEquation> of the big wreath product G∞, which are analogues of the Kerov–Olshanski–Vershik generalized regular representations of the infinite symmetric group. We derive an explicit formula for the characters of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11856_2025_2739_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="80" /> </InlineMediaObject> <EquationSource Format="TEX">$T_{z_{1}},\cdots,z_{k}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mi>T</mi> <mrow> <msub> <mi>z</mi> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>z</mi> <mrow> <mi>k</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>. The spectral measures of these representations are characterized in different ways. In particular, these spectral measures are associated with point processes whose correlation functions are explicitly computed. Thus, in representation-theoretic terms, the paper solves a natural problem of harmonic analysis for the big wreath products: our results describe the decomposition of <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11856_2025_2739_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="80" /> </InlineMediaObject> <EquationSource Format="TEX">$T_{z_{1}},\cdots,z_{k}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mi>T</mi> <mrow> <msub> <mi>z</mi> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>z</mi> <mrow> <mi>k</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> into irreducible components.

Generalized regular representations of big wreath products

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2025年2月17日

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Eugene Strahov

摘要

Let G be a finite group with k conjugacy classes, and S(∞) be the infinite symmetric group, i.e., the group of finite permutations of {1, 2, 3, …}. Then the wreath product G_∞ = G ∼ S (∞) of G with S (∞) (called the big wreath product) can be defined. The group G_∞ 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_∞. In particular, we construct an analogue ${\frak{S}}_{G}$ $S_{G}$ of the space of virtual permutations. We then formulate and prove a theorem characterizing all central probability measures on ${\frak{S}}_{G}$ $S_{G}$ . Next, we introduce generalized regular representations $\{T_{z_{1}},\cdots,z_{k} : z_{1}\in \mathbb{C},\cdots,z_{k}\in \mathbb{C}\}$ ${T_{z_{1}}, \dots, z_{k} : z_{1} \in C, \dots, z_{k} \in C}$ of the big wreath product G_∞, which are analogues of the Kerov–Olshanski–Vershik generalized regular representations of the infinite symmetric group. We derive an explicit formula for the characters of $T_{z_{1}},\cdots,z_{k}$ $T_{z_{1}}, \dots, z_{k}$ . The spectral measures of these representations are characterized in different ways. In particular, these spectral measures are associated with point processes whose correlation functions are explicitly computed. Thus, in representation-theoretic terms, the paper solves a natural problem of harmonic analysis for the big wreath products: our results describe the decomposition of $T_{z_{1}},\cdots,z_{k}$ $T_{z_{1}}, \dots, z_{k}$ into irreducible components.