## Journal of the Mathematical Society of Japan

- J. Math. Soc. Japan
- Volume 60, Number 4 (2008), 955 -1276

### Limits of characters of wreath products of a compact group $\bm{T}$ with the symmetric groups and characters of $\bm{\mathfrak{S}_\infty(T)}$ , II -- From a viewpoint of probability theory

Akihito HORA, Takeshi HIRAI, and Etsuko HIRAI

#### Abstract

This paper is the second part of our study on limiting behavior of characters of wreath products $\mathfrak{S}_n(T)$ of compact group $T$ as $n\to\infty$ and its connection with characters of $\mathfrak{S}_\infty(T)$ . Contrasted with the first part, which has a representation-theoretical flavor, the approach of this paper is based on probabilistic (or ergodic-theoretical) methods. We apply boundary theory for a fairly general branching graph of infinite valencies to wreath products of an arbitrary compact group $T$ . We show that any character of $\mathfrak{S}_\infty(T)$ is captured as a limit of normalized irreducible characters of $\mathfrak{S}_n(T)$ as $n\to\infty$ along a path on the branching graph of $\mathfrak{S}_\infty(T)$ . This yields reconstruction of an explicit character formula for $\mathfrak{S}_\infty(T)$ .

#### Article information

**Source**

J. Math. Soc. Japan Volume 60, Number 4 (2008), 1187-1217.

**Dates**

First available: 5 November 2008

**Permanent link to this document**

http://projecteuclid.org/euclid.jmsj/1225894038

**Digital Object Identifier**

doi:10.2969/jmsj/06041187

**Mathematical Reviews number (MathSciNet)**

MR2467875

**Zentralblatt MATH identifier**

05500756

**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]

**Keywords**

the infinite symmetric group character factor representation wreath product probabilistic method

#### Citation

HORA, Akihito; HIRAI, Takeshi; HIRAI, Etsuko. Limits of characters of wreath products S n ( T ) of a compact group T with the symmetric groups and characters of S ∞ ( T ) , II -- From a viewpoint of probability theory. Journal of the Mathematical Society of Japan 60 (2008), no. 4, 1187--1217. doi:10.2969/jmsj/06041187. http://projecteuclid.org/euclid.jmsj/1225894038.