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The cohomology rings of Hilbert schemes via Jack polynomials. (English) Zbl 1104.14002

Hurtubise, Jacques (ed.) et al., Algebraic structures and moduli spaces. Proceedings of the CRM workshop, Montréal, Canada, July 14-20, 2003. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3568-8/pbk). CRM Proceedings & Lecture Notes 38, 249-258 (2004).
Fundamental and deep connections have been developed in recent years between the geometry of Hilbert schemes \(X^{[n]}\) of points on a (quasi-)projective surface \(X\) and combinatorics of symmetric functions. Among distinguished classes of symmetric functions, let us mention the monomial symmetric functions, Schur polynomials, Jack polynomials (which depend on a Jack parameter), and Macdonald polynomials, etc. The monomial symmetric functions can be realized as certain ordinary cohomology classes of the Hilbert schemes associated to an embedded curve in a surface. Nakajima further showed that the Jack polynomials whose Jack parameter is a positive integer can be realized as certain \(\mathbb{T}\)-equivariant cohomology classes of the Hilbert schemes of points on the surface \(X(\gamma)\) which is the total space of the line bundle \({\mathcal{O}}_{\mathbb{P}^{1}}(-\gamma)\) over the complex projective line \({\mathbb{P}^{1}}\).
In the paper under review, generalizing earlier work of Nakajima and Vasserot, the authors study the (equivariant) cohomology rings of Hilbert schemes of certain toric surfaces and establish their connections to Fock space and Jack polynomials. In particular, they describe the equivariant and ordinary cohomology rings of the Hilbert schemes of points on the surface \(X(\gamma)\). They first show that the Jack polynomials can be realized in terms of certain \(\mathbb{T}\)-equivariant cohomology classes of the Hilbert schemes of points on the affine plane, and the Jack parameter comes from the ratio of the \(\mathbb{T}\)-weights on the two affine lines preserved by the \(\mathbb{T}\)-action. In addition, the authors study the \(\mathbb{T}\)-equivariant cohomology ring of \(X(\gamma)^{[n]}\) with respect to a certain \(\mathbb{T}\)-action. Finally, they note that the ordinary cohomology ring of the Hilbert scheme \(X(\gamma)^{[n]}\) can be shown to be isomorphic to the graded ring associated to a natural filtration on the ring \(H^{2n}_{\mathbb{T}}(X(\gamma)^{[n]})\). In this way, they obtain an algorithm for computing the ordinary cup product of cohomology classes in \(X(\gamma)^{[n]}\).
For the entire collection see [Zbl 1051.14002].

MSC:

14C05 Parametrization (Chow and Hilbert schemes)
05E05 Symmetric functions and generalizations
14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
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