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

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 $\mathfrak{S}_\infty(T)$ .