## Nagoya Mathematical Journal

- Nagoya Math. J.
- Volume 193 (2009), 1-93.

### Limits of characters of wreath products ${\mathfrak S}_{n}(T)$ of a compact group $T$ with the symmetric groups and characters of ${\mathfrak S}_{\infty}(T)$, I

Takeshi Hirai, Etsuko Hirai, and Akihito Hora

#### Abstract

In the first half of this paper, all the limits of irreducible characters of $G_{n} = {\mathfrak S}_{n}(T)$ as $n \to \infty$ are calculated. The set of all \emph{continuous} limit functions on $G = {\mathfrak S}_{\infty}(T)$ is exactly equal to the set of all characters of $G$ determined in [HH6]. We give a necessary and sufficient condition for a series of irreducible characters of $G_{n}$ to have a continuous limit and also such a condition to have a discontinuous limit. In the second half, we study the limits of characters of certain induced representations of $G_{n}$ which are usually reducible. The limits turn out to be characters of $G$, and we analyse which of irreducible components are responsible to these limits.

#### Article information

**Source**

Nagoya Math. J., Volume 193 (2009), 1-93.

**Dates**

First available in Project Euclid: 3 March 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.nmj/1236089981

**Mathematical Reviews number (MathSciNet)**

MR2502908

**Zentralblatt MATH identifier**

1182.22004

**Subjects**

Primary: 20C32: Representations of infinite symmetric groups

Secondary: 20C15: Ordinary representations and characters 20E22: Extensions, wreath products, and other compositions [See also 20J05] 22A25: Representations of general topological groups and semigroups 22C05: Compact groups 43A35: Positive definite functions on groups, semigroups, etc.

#### Citation

Hirai, Takeshi; Hirai, Etsuko; Hora, Akihito. Limits of characters of wreath products ${\mathfrak S}_{n}(T)$ of a compact group $T$ with the symmetric groups and characters of ${\mathfrak S}_{\infty}(T)$, I. Nagoya Math. J. 193 (2009), 1--93. https://projecteuclid.org/euclid.nmj/1236089981