The Farahat-Higman ring of wreath products and Hilbert schemes

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 $SL_2(C)$. The above results provide new insight to the cohomology rings of Hilbert schemes of points on a quasi-projective surface.