A description of characters on the infinite wreath product

2005/10/27 by A. V. Dudko, Dudko, A. V., N. I. Nessonov +1
Mathematics · #20C15 #20C32 #FOS: Mathematics #Representation Theory (math.RT) #math.RT #msc:20C15 #msc:20C32

paper · pdf · doi:10.48550/arxiv.math/0510597

33 pages. We receive the full description of the finite characters on the infinite wreath product

arxiv created 2005/10/27 · arxiv updated 2009/12/01

Abstract

Let \mathfrakS_∞ be the infinity permutation group and Γ an arbitrary group. Then \mathfrakS_∞ admits a natural action on Γ^∞ by automorphisms, so one can form a semidirect product Γ^∞\rtimes \mathfrakS_∞, known as the \it wreath product Γ\wr\mathfrakS_∞ of Γ by \mathfrakS_∞. We obtain a full description of unitary II₁-factor-representations of Γ\wr\mathfrakS_∞ in terms of finite characters of Γ. Our approach is based on extending Okounkov's classification method for admissible representations of \mathfrakS_∞×\mathfrakS_∞. Also, we discuss certain examples of representations of type II₁, where the \it modular operator of Tomita-Takesaki expresses naturally by the asymptotic operators, which are important in the characters-theory of infinite symmetric group.

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