\(K\)-theory of quiver varieties, \(q\)-Fock space and nonsymmetric Macdonald polynomials. (English) Zbl 1246.17020
There are two constructions of the level-\((0,1)\) irreducible representation of the quantum toroidal algebra of type \(A\). In the first (due to Nakajima and Varagnolo-Vasserot) the representation is constructed on the direct sum of the equivariant \(K\)-groups of the quiver varieties of type \(\widehat{A}\). In the second (due to Saito-Takemura-Uglov and Varagnolo-Vasserot) the representation is constructed on the \(q\)-deformed Fock space introduced by Kashiwara-Miwa-Stern.
In the paper an explicit isomorphism between these two constructions is given. To describe this isomorphism the author construct simultaneous eigenvectors on the \(q\)-Fock space using the nonsymmetric Macdonald polynomials. The isomorphism is given by attaching these vectors to the torus fixed points on the quiver varieties.
In the paper an explicit isomorphism between these two constructions is given. To describe this isomorphism the author construct simultaneous eigenvectors on the \(q\)-Fock space using the nonsymmetric Macdonald polynomials. The isomorphism is given by attaching these vectors to the torus fixed points on the quiver varieties.
Reviewer: Justyna Kosakowska (Toruń)
MSC:
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 33C52 | Orthogonal polynomials and functions associated with root systems |
| 14D21 | Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) |
| 16G20 | Representations of quivers and partially ordered sets |
Keywords:
representations of quantum toroidal algebras; q-Fock spaces; nonsymmetric Macdonald polynomials; K-theory of quiver representationsReferences:
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