Harmonic functions on the branching graph associated with the infinite wreath product of a compact group
Let \(T\) be a compact group, \(\mathfrak{S}_n\) the symmetric group of degree \(n\) and \(\mathfrak{S}_\infty\) the infinite symmetric group. The wreath products \(T\wr\mathfrak{S}_n\) and \(T\wr\mathfrak{S}_\infty\) are respectively denoted by \(\mathfrak{S}_n(T)\) and \(\mathfrak{S}_\infty(T)\). The paper under review is part of the authors' program of harmonic analysis on \(\mathfrak{S}_\infty(T)\). An important step in their project is a detailed study of the characters of this group, which was initiated in [\textit{T. Hirai} and \textit{E. Hirai}, Banach Cent. Publ. 73, 207--221 (2006; Zbl 1147.22005)] and [\textit{T. Hirai} and \textit{E. Hirai}, J. Math. Kyoto Univ. 47, No. 2, 269--320 (2007; Zbl 1176.20010)]. Further understanding of these representations was obtained in terms of positive definite functions in [\textit{T. Hirai} et al., Nagoya Math. J. 193, 1--93 (2009; Zbl 1182.22004)] and in terms of probability measures in [\textit{A. Hora} et al., J. Math. Soc. Japan 60, No. 4, 1187--1217 (2008; Zbl 1187.22001)]. The approach adopted in the article at hand consists in studying harmonic functions on the branching graph \(\mathbb{Y}(\hat{T})\) obtained from the inductive system of branching graphs associated with the groups \(\mathfrak{S}_n(T)\). The authors compute the Martin boundary of \(\mathbb{Y}(\hat{T})\) by using irreducible characters of \(\mathfrak{S}_n(T)\). They obtain an integral representation formula for harmonic functions on \(\mathbb{Y}(\hat{T})\) and combine with an asymptotic analysis of the irreducible characters of \(\mathfrak{S}_n(T)\) to characterize the minimal Martin boundary of \(\mathbb{Y}(\hat{T})\). A classification of the minimal nonnegative harmonic functions on \(\mathbb{Y}(\hat{T})\) is obtained, which implies a classification of the characters of \(\mathfrak{S}_\infty(T)\). Although the latter classification result was already known, due to the aforementioned work of the authors and E. Hirai, this article is self-contained and adopts a transverse approach, yielding constructions and proofs of independent interest, as well as complementary insight on the representation theory of \(\mathfrak{S}_\infty(T)\).
scientific article; zbMATH DE number 6381915