Quiver varieties and Hilbert schemes

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In this note we give an explicit geometric description of some of the Nakajima's quiver varieties. More precisely, we show that the Γ-equivariant Hilbert scheme XΓ[ⁿ] and the Hilbert scheme XΓ[ⁿ] (where X=², Γ⊂ SL(²) is a finite subgroup, and XΓ is a minimal resolution of X/Γ) are quiver varieties for the affine Dynkin graph, corresponding to Γ via the McKay correspondence, the same dimension vectors, but different parameters ζ (for earlier results in this direction see [4, 12, 13]). In particular, it follows that the varieties XΓ[ⁿ] and XΓ[ⁿ] are diffeomorphic. Computing their cohomology (in the case Γ=/d) via the fixed points of (×)-action we deduce the following combinatorial identity: the number UCY(n,d) of uniformly coloured in d colours Young diagrams consisting of nd boxes coincides with the number CY(n,d) of collections of d Young diagrams with the total number of boxes equal to n.