Quiver varieties and Hilbert schemes

In this note we give an explicit geometric description of some of the Nakajima's quiver varieties. More precisely, if $X=\mathbb C^2$, $\Gamma\subset{\rm SL}(\mathbb C^2)$ is a finite subgroup, and $X_\Gamma$ is a minimal resolution of $X/\Gamma$, we show that $X^{\Gamma[n]}$ (the $\Gamma$-equivariant Hilbert scheme of $X$), and $X_{\Gamma}^{[n]}$ (the Hilbert scheme of $X_{\Gamma}$) are quiver varieties for the affine Dynkin graph corresponding to $\Gamma$ via the McKay correspondence with the same dimension vectors but different parameters $\zeta$ (for earlier results in this direction see works by M. Haiman, M. Varagnolo and E. Vasserot, and W. Wang). In particular, it follows that the varieties $X^{\Gamma[n]}$ and $X_{\Gamma}^{[n]}$ are diffeomorphic. Computing their cohomology (in the case $\Gamma=\mathbb Z/d\mathbb Z$) via the fixed points of a $(\mathbb C^*\times\mathbb C^*)$-action we deduce the following combinatorial identity: the number $UCY(n,d)$ of Young diagrams consisting of $nd$ boxes and uniformly colored in $d$ colors equals the number $CY(n,d)$ of collections of $d$ Young diagrams with the total number of boxes equal to $n$.