# Hilbert schemes, Hecke algebras and the Calogero-Sutherland system

@article{Costello2003HilbertSH, title={Hilbert schemes, Hecke algebras and the Calogero-Sutherland system}, author={Kevin J. Costello and Ian Grojnowski}, journal={arXiv: Algebraic Geometry}, year={2003} }

We describe the ring structure of the cohomology of the Hilbert scheme of points for a smooth surface X. When X is C 2 , this was done in [13, 21] by realising this ring as a degeneration of the center of CSn. When the canonical class KX = 0, [14] extended this result by defining an algebra structure on H � ({(x, g) ∈ X n × Sn | gx = x}); the Sn-invariants of this algebra is the desired ring. But when KX 6 0 it seems no such algebra can exist. A completely different approach is needed. Instead…

## 18 Citations

### Two point extremal Gromov–Witten invariants of Hilbert schemes of points on surfaces

- Mathematics
- 2007

Given an algebraic surface X, the Hilbert scheme X[n] of n-points on X admits a contraction morphism to the n-fold symmetric product X(n) with the extremal ray generated by a class βn of a rational…

### The Chow ring of punctual Hilbert schemes on toric surfaces

- Mathematics
- 2005

Let X be a smooth projective toric surface, and ${\mathbb H}^d(X)$ the Hilbert scheme parametrizing the length d zero-dimensional subschemes of X. We compute the rational Chow ring $A^*({\mathbb…

### On the Instanton R-matrix

- MathematicsCommunications in Mathematical Physics
- 2016

A torus action on a symplectic variety allows one to construct solutions to the quantum Yang-Baxter equations (R-matrices). For a torus action on cotangent bundles over flag varieties the resulting…

### On the Instanton R-matrix

- Mathematics
- 2013

A torus action on a symplectic variety allows one to construct solutions to the quantum Yang-Baxter equations (R-matrices). For a torus action on cotangent bundles over flag varieties the resulting…

### Integral generators for the cohomology ring of moduli spaces of sheaves over Poisson surfaces

- Mathematics
- 2004

### Intersection theory on punctual Hilbert schemes and graded Hilbert schemes

- Mathematics
- 2010

The rational Chow ring A?(S[n],Q) of the Hilbert scheme S[n] parametrising the length n zero-dimensional subschemes of a toric surface S can be described with the help of equivariant techniques. In…

### Generating Series of the Poincaré Polynomials of Quasihomogeneous Hilbert Schemes

- Mathematics
- 2012

In this paper we prove that the generating series of the Poincare polynomials of quasihomogeneous Hilbert schemes of points in the plane has a beautiful decomposition into an infinite product. We…

### Quantum cohomology of the Hilbert scheme of points in the plane

- Mathematics
- 2004

We determine the ring structure of the equivariant quantum cohomology of the Hilbert scheme of points of ℂ2. The operator of quantum multiplication by the divisor class is a nonstationary deformation…

### Affine zigzag algebras and imaginary strata for KLR algebras

- MathematicsTransactions of the American Mathematical Society
- 2018

KLR algebras of affine
ADE
\texttt {ADE}
types are known to be properly stratified if the characteristic of the ground field is greater than some explicit bound. Understanding the…

### Affine wreath product algebras

- Mathematics
- 2017

We study the structure and representation theory of affine wreath product algebras and their cyclotomic quotients. These algebras, which appear naturally in Heisenberg categorification,…

## References

SHOWING 1-10 OF 22 REFERENCES

### Heisenberg algebra and Hilbert schemes of points on projective surfaces

- Mathematics
- 1995

The purpose of this paper is to throw a bridge between two seemingly unrelated subjects. One is the Hilbert scheme of points on projective surfaces, which has been intensively studied by various…

### Instantons and affine algebras I: The Hilbert scheme and vertex operators

- Mathematics
- 1995

This is the first in a series of papers which describe the action of an affine Lie algebra with central charge $n$ on the moduli space of $U(n)$-instantons on a four manifold $X$. This generalises…

### Cherednik algebras and differential operators on quasi-invariants

- Mathematics
- 2001

We develop representation theory of the rational Cherednik algebra H associated to a finite Coxeter group W in a vector space h. It is applied to show that, for integral values of parameter `c', the…

### The Betti numbers of the Hilbert scheme of points on a smooth projective surface

- Mathematics
- 1990

Several authors have been interested in the Hilbert scheme SPq:=Hilb"(S) parametrizing finite subschemes of length n on a smooth projective surface S. In EF 1] and I-F 2] Fogarty shows that S tnj is…

### Hilbert schemes and W algebras

- Mathematics
- 2001

We construct geometrically the generating fields of a W algebra which acts irreducibly on the direct sum of the cohomology rings of the Hilbert schemes of n points on a projective surface for all n.…

### Chern classes of tautological sheaves on Hilbert schemes of points on surfaces

- Mathematics
- 1999

Abstract. We give an algorithmic description of the action of the Chern classes of tautological bundles on the cohomology of Hilbert schemes of points on a smooth surface within the framework of…

### Lectures on Knizhnik-Zamolodchikov equations and Hecke algebras

- Mathematics
- 1998

This paper is the course of lectures delivered by the first author in Kyoto in 1996-97 and recorded by the others. We tried to follow closely the notes of the lectures not yielding to the temptation…

### MATRIX THEORY, HILBERT SCHEME AND INTEGRABLE SYSTEM

- Mathematics
- 1998

We give a reinterpretation of the matrix theory discussed by Moore, Nekrasov and Shatashivili (MNS) in terms of the second quantized operators which describes the homology class of the Hilbert scheme…

### The cup product of the Hilbert scheme for K3 surfaces

- Mathematics
- 2000

To any graded Frobenius algebra A we associate a sequence of graded Frobenius algebras A^[n] in such a way that for any smooth projective surface X with trivial canonical divisor there is a canonical…

### Symmetric groups and the cup product on the cohomology of Hilbert schemes

- Mathematics
- 2000

Let C(Sn) be the Z-module of integer valued class functions on the symmetric group Sn. We introduce a graded version of the con- volution product on C(Sn) and show that there is a degree preserving…