2002/03/31 by Weiqiang Wang
Mathematics · #math.QA #math.RT
published as Proc. London Math. Soc. 88 (2004), 381--404. · 25 pages, latex, minor changes, to appear in Proc. London Math. Soc
arxiv created 2003/02/18 · arxiv updated 2009/11/30
The Jucys-Murphy elements for wreath products Gn associated to any finite group G are introduced and they play an important role in our study on the connections between class algebras of Gn for all n and vertex algebras. We construct an action of (a variant of) the W1+∞ algebra acting irreducibly on the direct sum RG of the class algebras of Gn for all n in a group theoretic manner. We establish various relations between convolution operators using JM elements and Heisenberg algebra operators acting on RG. As applications, we obtain two distinct sets of algebra generators for the class algebra of Gn and establish various stability results concerning products of normalized conjugacy classes of Gn and the power sums of Jucys-Murphy elements etc. We introduce a stable algebra which encodes the class algebra structures of Gn 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.