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The \({\mathbb {A}}_{q,t}\) algebra and parabolic flag Hilbert schemes


The earlier work of the first and the third name authors introduced the algebra \({\mathbb {A}}_{q,t}\) and its polynomial representation. In this paper we construct an action of this algebra on the equivariant K-theory of certain smooth strata in the flag Hilbert scheme of points on the plane. In this presentation, the fixed points of the torus action correspond to generalized Macdonald polynomials, and the matrix elements of the operators have an explicit presentation.

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    A categorically inclined reader can view our algebras as categories with object set \({\mathbb {Z}}_{\ge 0}\). Then a representation of a category is a simply a functor to the category of vector spaces.


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The authors would like to thank Mikhail Bershtein, Andrei Neguț and Monica Vazirani for the useful discussions. The work of E. G. was partially supported by the NSF grants DMS-1559338 and DMS-1700814, Russian Academic Excellence Project 5-100 and RSF-16-11-10160. The work of A. M. was supported by the Advanced Grant “Arithmetic and Physics of Higgs moduli spaces” No. 320593 of the European Research Council and by the START-Project Y963-N35 of the Austrian Science Fund.

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Correspondence to Erik Carlsson.

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Carlsson, E., Gorsky, E. & Mellit, A. The \({\mathbb {A}}_{q,t}\) algebra and parabolic flag Hilbert schemes. Math. Ann. (2019).

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