0 An explicit construction of the McKay correspondence for A - Hilb C 3

For a finite Abelian subgroup A ⊂ SL(3,C), Ito and Nakajima [IN00] established an isomorphism between the K-theory of Nakamura’s A-Hilbert scheme A-Hilb C3 and the representation ring of A. This leads to a basis of the rational cohomology of A-Hilb C3 in one-to-one correspondence with the irreducible representations of A. In this paper we construct an explicit basis of the integral cohomology of A-Hilb C3 in one-to-one correspondence with the irreducible representations of A. Our approach is elementary, leading to some lovely pictures of toric fans decorated with characters of A.