Equivariant \(K\)-theory of Hilbert schemes via shuffle algebra. (English) Zbl 1242.14006

The authors construct an action of Ding-Iohara and shuffle algebras on the sum \(M\) of localized equivariant \(K\)-groups of Hilbert schemes of points on \(\mathbb{C}^2\). This yields an action of the Heisenberg algebra. Further, the authors construct an isomorphism between \(M\) and the Fock space which takes elements of the normalized fixed point basis to Macdonald polynomials and leads to a realization of the above action via Pieri formulae.


14C05 Parametrization (Chow and Hilbert schemes)
19E08 \(K\)-theory of schemes
17B69 Vertex operators; vertex operator algebras and related structures
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[1] A. Braverman and M. Finkelberg, Finite difference quantum Toda lattice via equivariant K-theory , Transform. Groups 10 (2005), 363-386. · Zbl 1122.17008 · doi:10.1007/s00031-005-0402-4
[2] B. Feigin, E. Feigin, M. Jimbo, T. Miwa, and E. Mukhin, Quantum continuous gl \infty : Semiinfinite construction of representations , Kyoto J. Math. 51 (2011), 365-392. · Zbl 1278.17013 · doi:10.1215/21562261-1214384
[3] B. Feigin, M. Finkelberg, A. Negut, and L. Rybnikov, Yangians and cohomology rings of Laumon spaces , Select Math. (N.S.) 17 (2011), 573-607. · Zbl 1260.14015
[4] B. Feigin, K. Hashizume, J. Shiraishi, and S. Yanagida, A commutative algebra on degenerate \Bbb C\Bbb P 1 and Macdonald polynomials , J. Math. Phys. 50 (2009), no. 095215. · Zbl 1248.33034
[5] B. Feigin and A. Odesskii, “Vector bundles on elliptic curve and Sklyanin algebras” in Topics in Quantum Groups and Finite-Type Invariants , Amer. Math. Soc. Transl. Ser. 2 185 , Amer. Math. Soc., Providence, 1998, 65-84.
[6] W. Li, Z. Qin, and W. Wang, “The cohomology rings of Hilbert schemes via Jack polynomials” in Algebraic Structures and Moduli Spaces , CRM Proc. Lecture Notes 38 , Amer. Math. Soc., Providence, 2004, 249-258. · Zbl 1104.14002
[7] I. Macdonald, Symmetric Functions and Hall Polynomials , 2nd ed., Oxford Math. Monogr., Oxford Univ. Press, New York, 1995. · Zbl 0824.05059
[8] H. Nakajima, Lectures on Hilbert Schemes of Points on Surfaces , Univ. Lecture Ser. 18 , Amer. Math. Soc., Providence, 1999. · Zbl 0949.14001
[9] O. Schiffmann, Drinfeld realization of the elliptic Hall algebra , J. Algebr. Comb. (2011) DOI 10.1007/s10801-011-0302-8. · Zbl 1271.17010
[10] O. Schiffmann and E. Vasserot, The elliptic Hall algebra and the equivariant K-theory of the Hilbert scheme of \(\mathbb{A}^{2}\), · Zbl 1290.19001
[11] A. Tsymbaliuk, Quantum affine Gelfand-Tsetlin bases and quantum toroidal algebra via K-theory of affine Laumon spaces , Selecta Math. (N.S.) 16 (2010), 173-200. · Zbl 1284.17011 · doi:10.1007/s00029-009-0013-3
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