Kyoto Journal of Mathematics

Equivariant $K$-theory of Hilbert schemes via shuffle algebra

Abstract

In this paper we construct the action of Ding-Iohara and shuffle algebras on the sum of localized equivariant $K$-groups of Hilbert schemes of points on $\mathbb{C}^{2}$. We show that commutative elements $K_{i}$ of shuffle algebra act through vertex operators over the positive part $\{\mathfrak{h}_{i}\}_{i\textgreater 0}$ of the Heisenberg algebra in these $K$-groups. Hence we get an action of Heisenberg algebra itself. Finally, we normalize the basis of the structure sheaves of fixed points in such a way that it corresponds to the basis of Macdonald polynomials in the Fock space $\mathbb{C}[\mathfrak{h}_{1},\mathfrak{h}_{2},\ldots]$.

Article information

Source
Kyoto J. Math., Volume 51, Number 4 (2011), 831-854.

Dates
First available in Project Euclid: 10 November 2011

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1320936734

Digital Object Identifier
doi:10.1215/21562261-1424875

Mathematical Reviews number (MathSciNet)
MR2854154

Zentralblatt MATH identifier
1242.14006

Subjects
Primary: 16Gxx: Representation theory of rings and algebras

Citation

Feigin, B. L.; Tsymbaliuk, A. I. Equivariant $K$ -theory of Hilbert schemes via shuffle algebra. Kyoto J. Math. 51 (2011), no. 4, 831--854. doi:10.1215/21562261-1424875. https://projecteuclid.org/euclid.kjm/1320936734

References

• [1] A. Braverman and M. Finkelberg, Finite difference quantum Toda lattice via equivariant K-theory, Transform. Groups 10 (2005), 363–386.
• [2] B. Feigin, E. Feigin, M. Jimbo, T. Miwa, and E. Mukhin, Quantum continuous gl: Semiinfinite construction of representations, Kyoto J. Math. 51 (2011), 365–392.
• [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.
• [4] B. Feigin, K. Hashizume, J. Shiraishi, and S. Yanagida, A commutative algebra on degenerate ℂℙ1 and Macdonald polynomials, J. Math. Phys. 50 (2009), no. 095215.
• [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.
• [7] I. Macdonald, Symmetric Functions and Hall Polynomials, 2nd ed., Oxford Math. Monogr., Oxford Univ. Press, New York, 1995.
• [8] H. Nakajima, Lectures on Hilbert Schemes of Points on Surfaces, Univ. Lecture Ser. 18, Amer. Math. Soc., Providence, 1999.
• [9] O. Schiffmann, Drinfeld realization of the elliptic Hall algebra, J. Algebr. Comb. (2011) DOI 10.1007/s10801-011-0302-8.
• [10] O. Schiffmann and E. Vasserot, The elliptic Hall algebra and the equivariant K-theory of the Hilbert scheme of $\mathbb{A}^{2}$, arXiv:0905.2555v2 [math.QA]
• [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.