The Farahat-higman Ring of Wreath Products and Hilbert Schemes

We study the structure constants of the class algebra RZ(Γn) of the wreath products Γn associated to an arbitrary finite group Γ with respect to a basis provided by the conjugacy classes. A suitable filtration on the RZ(Γn) gives rise to the rings GΓ(n) with non-negative integer structure constants independent of n, which are then encoded in a single (Farahat-Higman) ring GΓ. We establish various structure results of the rings GΓ and GΓ(n), where the real conjugacy classes of Γ come to play a distinguished role. If Γ is a subgroup of SL2(C), the ring C ⊗Z GΓ(n) is identified with the cohomology ring of the Hilbert schemes X [n] of n points when X is the minimal resolution C//Γ of a simple singularity C/Γ; the set of real conjugacy classes of Γ admits an elegant interpretation via the dual McKay correspondence. Our results on the Farahat-Higman ring of wreath products provide new insight to the cohomology rings of (C//Γ) and further lead to predictions on the cohomology rings of X , most notably for a quasi-projective surface X .