## Kyoto Journal of Mathematics

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

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

Akihito Hora and Takeshi Hirai

#### Abstract

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