The Farahat-Higman ring of wreath products and Hilbert schemes

2002/05/31 by Weiqiang Wang · 1 citation
Mathematics · #math.QA #math.AG

paper · pdf

published as Adv. in Math. 187 (2004), 417--446. · latex, abstract/introduction modified, to appear in Advances in Math

arxiv created 2003/09/16 · arxiv updated 2009/11/30

Abstract

We study the structure constants of the class algebra R_Z(G_n) of the wreath products G_n associated to an arbitrary finite group G with respect to the basis of conjugacy classes. We show that a suitable filtration on R_Z(G_n) gives rise to the graded ring \mathcal G_G(n) with non-negative integer structure constants independent of n (some of which are computed), which are then encoded in a Farahat-Higman ring \mathcal G_G. The real conjugacy classes of G come to play a distinguished role, and is treated in detail in the case when G is a subgroup of SL₂(C). The above results provide new insight to the cohomology rings of Hilbert schemes of points on a quasi-projective surface.

Cited by

• Stability of the centers of group algebras of general affine groups GA_n(q)

Related

• Algebraic structures behind Hilbert schemes and wreath products
• Ideals of the cohomology rings of Hilbert schemes and their applications
• Stable Centres II: Finite Classical Groups
• Universal rings arising in geometry and group theory
• Symmetric groups and the cup product on the cohomology of Hilbert schemes