Algebraic structures behind Hilbert schemes and wreath products

Wang, Weiqiang

doi:10.1090/conm/297/05102

Cited by 12 publications

(14 citation statements)

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“…This fact is well known, see e.g. [13], [12]; we include a proof only for the sake of completeness. We also prove that for ζ R ∈ C − (n) the quiver variety M (0,ζ R ) (nv 0 , w 0 ) is isomorphic to the Hilbert scheme of points on X Γ .…”

Section: Introductionmentioning

confidence: 69%

Quiver Varieties and Hilbert Schemes

Kuznetsov¹

2007

MMJ

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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 X Γ[n] and the Hilbert scheme X[n] Γ (where X = C 2 , Γ ⊂ SL(C 2 ) is a finite subgroup, and X Γ 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 X Γ[n] and X [n] Γ are diffeomorphic. Computing their cohomology (in the case Γ = Z/dZ) via the fixed points of (C * × C * )-action we deduce the following combinatorial identity: the number U CY (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. I was partially supported by RFFI grants 99-01-01144 and 99-01-01204.

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Section: Introductionmentioning

confidence: 69%

Quiver Varieties and Hilbert Schemes

Kuznetsov¹

2007

MMJ

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“…There is an action on R Γ (cf. [FJW,Wa1]) of a Heisenberg algebra which is generated by a n (γ i ), n ∈ Z, 0 ≤ i ≤ r − 1 with the commutation relation:…”

Section: Fixed Point Classes and Bilinear Forms Letmentioning

confidence: 99%

“…[n] of points on a surface X (see [Na2,Le,LQW1,LQW2,Ru,Vas,Wa1] and the references therein). To a large extent, the construction of Heisenberg algebra by Nakajima [Na1] (also cf.…”

Section: Introductionmentioning

confidence: 99%

“…When X = C 2 //Γ is the minimal resolution of C 2 /Γ, where Γ is a finite subgroup of SL 2 (C), we have the following resolution of singularities [Wa1] π n : (C 2 //Γ) [n] −→ C 2n /Γ n obtained by the composition (C 2 //Γ) [n] → (C 2 //Γ) n /S n → (C 2 /Γ) n /S n ∼ = C 2n /Γ n . Here Γ n := Γ n ⋊ S n is the wreath product.…”

Section: Introductionmentioning

confidence: 99%

“…This was the starting point on why Hilbert schemes have something to do with wreath products (cf. [Wa1] and the references therein).…”

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confidence: 99%

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Hilbert schemes of points on the minimal resolution and soliton equations

Qin¹,

Wang²

2007

Lie Algebras, Vertex Operator Algebras and Their Applications

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Abstract. The equivariant and ordinary cohomology rings of Hilbert schemes of points on the minimal resolution C 2 //Γ for cyclic Γ are studied using vertex operator technique, and connections between these rings and the class algebras of wreath products are explicitly established. We further show that certain generating functions of equivariant intersection numbers on the Hilbert schemes and related moduli spaces of sheaves on C 2 //Γ are τ -functions of 2-Toda hierarchies.

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