Quiver varieties and Hilbert schemes
Quiver varieties and Hilbert schemes
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DOI：
10.17323/1609-4514-2007-7-4-673-697
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发表时间：
2001-11
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通讯作者：
A. Kuznetsov
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作者：
A. Kuznetsov
本文给出了一些Nakajima的类簇的明确的几何描述。更精确地说，我们证明了$\Gamma$-等变Hilbert格式$X^{\Gamma[n]}$和Hilbert格式$X_\Gamma^{[n]}$（其中$X=\C^2 $，$\Gamma\subset SL（\C^2）$是有限子群，$X_\Gamma$是$X/\Gamma$的最小分解）是仿射Dynkin图的一个簇，通过McKay对应对应于$\Gamma$，相同的维度向量，但不同的参数$\zeta$（关于这个方向的早期结果，请参见[4，12，13]）。特别地，它遵循变种$X^{\Gamma[n]}$和$X_\Gamma^{[n]}$是同构的。通过$（\C^*\times\C^*）$-作用的不动点计算它们的上同调（在$\Gamma=\Z/d\Z$的情况下），我们推导出以下组合恒等式：由nd个盒子组成的d个颜色均匀着色的Young图的数目$UCY（n，d）$与盒子总数等于n的d个Young图的集合的数目$CY（n，d）$重合。
In this note we give an explicit geometric description of some of the Nakajima's quiver varieties. More precisely, we show that the $\Gamma$-equivariant Hilbert scheme $X^{\Gamma[n]}$ and the Hilbert scheme $X_\Gamma^{[n]}$ (where $X=\C^2$, $\Gamma\subset SL(\C^2)$ is a finite subgroup, and $X_\Gamma$ is a minimal resolution of $X/\Gamma$) are quiver varieties for the affine Dynkin graph, corresponding to $\Gamma$ via the McKay correspondence, the same dimension vectors, but different parameters $\zeta$ (for earlier results in this direction see [4, 12, 13]). In particular, it follows that the varieties $X^{\Gamma[n]}$ and $X_\Gamma^{[n]}$ are diffeomorphic. Computing their cohomology (in the case $\Gamma=\Z/d\Z$) via the fixed points of $(\C^*\times\C^*)$-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.