Quiver varieties and Hilbert schemes

By Alexander Kuznetsov. 2001

60 citations recorded by the source corpus.

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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d39f52fd78c7ce0531544c0ebe3c48f70de6f7da

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In Discrete Mathematics and Combinatorics

1. Pavel Etingof
2. D. Kaledin
3. H. Nakajima
4. Y. Berest
5. V. Ginzburg
6. D. Hernandez
7. N. Nekrasov
8. I. Gordon

In Geometry and Topology

1. V. Ginzburg
2. Pavel Etingof
3. I. Gordon
4. Y. Berest
5. I. Losev
6. D. Kaledin
7. H. Nakajima
8. Michel van den Bergh

In Mathematical Physics

1. V. Ginzburg
2. Pavel Etingof
3. I. Losev
4. I. Gordon
5. Y. Berest
6. D. Kaledin
7. H. Nakajima
8. Michel van den Bergh

Citations in the corpus

• Geometric Construction of Representations of Affine Algebras (2002, background): “mer’s construction [20]. The precise explanation was given in [29, Chapter 4]. Recently higher dimensional M(v) attract attention in connection with the McKay correspondence for wreath products Γ ≀Sn [44, 22, 15]. These M(v) are diffeomorphic to the Hilbert schemes of points on the minimal resolution”
• Cherednik algebras and differential operators on quasi-invariants (2001, background): “s an interesting interpretation in terms of the Calogero-Moser space Mn, cf. [Wi]. It is known that fixed points of a natural C∗-action on Mn are labelled by Young diagrams, cf. e.g. [EG, Prop. 4.16], [Ku]. Let (Xτ,Zτ) denote a pair of n×n-matrices in the conjugacy class corresponding to the fixed poi”
• Poisson deformations of symplectic quotient singularities (2002, result): “= h⊗ C2, via the action on the first factor, by symplectic automorphisms. For types An, Bn, Cn the quotient V/Gis known to admit a crepant resolution of singularities (of Hilbert scheme type, see e.g. [Ku]). Using the results of I. Gordon [Go], we resolve the existence question for other Dynkin graph”