The Farahat–Higman ring of wreath products and Hilbert schemes

Abstract

We study the structure constants of the class algebra R Z (Γ n) of the wreath products Γ n associated to an arbitrary finite group Γ with respect to the basis of conjugacy classes. We show that a suitable filtration on R Z (Γ n) gives rise to the graded ring G Γ(n) with non-negative integer structure constants independent of n (some of which are computed), which are then encoded in a Farahat–Higman ring G Γ . The real conjugacy classes of Γ come to play a distinguished role and are treated in detail in the case when Γ is a subgroup of SL 2( C) . The above results provide new insight to the cohomology rings of Hilbert schemes of points on a quasi-projective surface X.