Symmetric groups and the cup product on the cohomology of Hilbert schemes

Abstract

Let $\mathscr {C}(S_n)$ be the $\mathbb {Z}$-module of integer-valued class functions on the symmetric group $S_n)$. We introduce a graded version of the convolution product on $\mathscr {C}(S_n)$, and we show that there is a degree-preserving ring isomorphism $\mathscr {C}(S_n)\longrightarrow H^\ast({\rm Hilb}^n(\mathbb {A_C}^2);\mathbb {Z})$ to the cohomology of the Hilbert scheme of points in the complex affine plane.