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

Let C(Sₙ) be the Z-module of integer-valued class functions on the symmetric group Sₙ). We introduce a graded version of the convolution product on C(Sₙ), and we show that there is a degree-preserving ring isomorphism C(Sₙ) H⁽ Hilbⁿ(AC²);Z) to the cohomology of the Hilbert scheme of points in the complex affine plane.