Mark Haiman - The McKay correspondence and the n! conjecture

The McKay correspondence is a remarkable conjecture asserting that if G is a finite group of linear endomorphisms with determinatn 1 of a complex vector space V, and X is a special type (called crepant) of resolution of singularities of the orbit space V/G, then the Betti numbers of X sum to the number of conjugacy classes of G. As a step toward explaining the McKay correspondence, Nakamura has proposed that a space known as the G-Hilbert scheme should be a crepant resolution of V/G whenever one exists. When Gis the symmetric group acting on the direct sum of two copies of its natural representation, Nakamura's conjecture is equivalent to the ``n! conjecture'' of Garsia and myself.