Vertex Algebras and the Class Algebras of Wreath Products

The Jucys-Murphy elements for wreath products Γn associated to any finite group Γ are introduced and they play an important role in our study on the connections between class algebras of Γn for all n and vertex algebras. We construct an action of (a variant of) the W1+∞ algebra acting irreducibly on the direct sum RΓ of the class algebras of Γn for all n in a group theoretic manner. We establish various relations between convolution operators using JM elements and Heisenberg algebra operators acting on RΓ. As applications, we obtain two distinct sets of algebra generators for the class algebra of Γn and establish various stability results concerning products of normalized conjugacy classes of Γn and the power sums of Jucys-Murphy elements etc. We introduce a stable algebra which encodes the class algebra structures of Γn for all n, whose structure constants are shown to be non-negative integers. In the symmetric group case (i.e. Γ is trivial), we recover and strengthen in a uniform approach various results of Lascoux-Thibon, Kerov-Olshanski, and Farahat-Higman, etc.