Thoma's theorem and representations of the infinite bisymmetric group (Q1896954)

Let \(S(\infty)\) be the group of permutations, with finite support, of the set \(\mathbb{N} = \{1,2,\dots \}\) of positive integers. Thoma's theorem describes all characters of this group. Thoma's original proof was based on a rather nontrivial argument involving entire functions. Later Vershik and Kerov found another way of proving the theorem based on the approximation of the characters of the group \(S(\infty)\) by characters of finite symmetric groups. In this paper we prove Thoma's theorem by developing the ideas and the method of \textit{G. I. Ol'shanskij} [Algebra Anal. 1, No. 4, 178-209 (1989; Zbl 0731.20009)]. Our method also makes it possible to substantially simplify Thoma's original proof. Along with these results, we will obtain a classification of an important class of representations of the bisymmetric group; this class was introduced in [loc. cit.]. All the constructions can be carried out simultaneously for the other \((G,K)\) pairs in [loc. cit.].