Kyoto Journal of Mathematics

Harmonic functions on the branching graph associated with the infinite wreath product of a compact group

Akihito Hora and Takeshi Hirai

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Abstract

A detailed study of the characters of S(T), the wreath product of compact group T with the infinite symmetric group S, is indispensable for harmonic analysis on this big group. In preceding works, we investigated limiting behavior of characters of the finite wreath product Sn(T) as n and its connection with characters of S(T). This paper takes a dual approach to these problems. We study harmonic functions on Y(T^), the branching graph of the inductive system of Sn(T)’s, and give a classification of the minimal nonnegative harmonic functions on it. This immediately implies a classification of the characters of S(T), which is a logically independent proof of the one obtained in earlier works. We obtain explicit formulas for minimal nonnegative harmonic functions on Y(T^) and Martin integral expressions for harmonic functions.

Article information

Source
Kyoto J. Math., Volume 54, Number 4 (2014), 775-817.

Dates
First available in Project Euclid: 5 November 2014

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1415196156

Digital Object Identifier
doi:10.1215/21562261-2801822

Mathematical Reviews number (MathSciNet)
MR3276417

Zentralblatt MATH identifier
1306.22002

Subjects
Primary: 20C32: Representations of infinite symmetric groups
Secondary: 20P05: Probabilistic methods in group theory [See also 60Bxx] 20E22: Extensions, wreath products, and other compositions [See also 20J05]

Citation

Hora, Akihito; Hirai, Takeshi. Harmonic functions on the branching graph associated with the infinite wreath product of a compact group. Kyoto J. Math. 54 (2014), no. 4, 775--817. doi:10.1215/21562261-2801822. https://projecteuclid.org/euclid.kjm/1415196156


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