Vertex algebras and the class algebras of wreath products

Abstract

The Jucys-Murphy elements for wreath products G_n associated to any finite group G are introduced and they play an important role in our study on the connections between class algebras of G_n for all n and vertex algebras. We construct an action of (a variant of) the W_{1+\infty} algebra acting irreducibly on the direct sum R_G of the class algebras of G_n for all n in a group theoretic manner. We establish various relations between convolution operators using JM elements and Heisenberg algebra operators acting on R_G. As applications, we obtain two distinct sets of algebra generators for the class algebra of G_n and establish various stability results concerning products of normalized conjugacy classes of G_n and the power sums of Jucys-Murphy elements etc. We introduce a stable algebra which encodes the class algebra structures of G_n for all n, whose structure constants are shown to be non-negative integers. In the symmetric group case (i.e. G is trivial), we recover and strengthen in a uniform approach various results of Lascoux-Thibon, Kerov-Olshanski, and Farahat-Higman, etc.