Abstract
We consider the gl N -invariant Calogero—Sutherland models with N = 1, 2, 3, … in the framework of symmetric polynomials. The Hamiltonian of any such model admits a distinguished orthogonal eigenbasis characterized as the union of Yangian Gelfand—Zetlin bases of irreducible components with respect to the Yangian action on the space of states. We construct an isomorphism from the space of states into the space of symmetric Laurent polynomials that maps the eigenbasis into the basis of gl N - Jack polynomials. These polynomials are defined as specializations of Macdonald polynomials where both parameters approach an Nth primitive root of unity. As an application of this isomorphism we compute two-point dynamical spin-density and density correlation functions in the gl 2-invariant Calogero-Sutherland model at integer values of the coupling constant.
Keywords
- Symmetric Polynomial
- Laurent Polynomial
- Macdonald Polynomial
- Jack Polynomial
- Density Correlation Function
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References
T. H. Baker and P. J. Forrester, Nonsymmetric Jack polynomials and integral kernels, Duke Math. J. 95 (1998), No. 1, 1–50, q-alg/9612003.
D. Bernard, M. Gaudin, F. D. M. Haldane, and V. Pasquier, Yang-Baxter equation in long-range interacting systems, J. Phys. A 26 (1993), No. 20, 5219–5236, hep-th/9301084.
I. Cherednik, Integration of quantum many-body problems by affine Knizhnik-Zamolodchikov equations, Adv. Math. 106 (1994), No. 1, 65–95.
I. Cherednik, Double affine Hecke algebras and Macdonald’s conjectures, Ann. of Math. 141 (1995), No. 1, 191–216.
I. Cherednik, Nonsymmetric Macdonald polynomials, Internat. Math. Res. Notices (1995), No. 10, 483–515.
I. V. Cherednik, A new interpretation of Gelfand-Tzetlin bases, Duke Math. J. 54 (1987), No. 2, 563–577.
V. G. Drinfeld, Degenerate affine Hecke algebras and Yangians, Funct. Anal. Appl. 20 (1986), No. 1, 62–64.
V. G. Drinfeld, Quantum groups, Proc. International Congress of Mathematicians (Berkeley, CA, 1986) (A. M. Gleason, ed.), Amer. Math. Soc, Providence, RI, 1987, pp. 798–820.
V. G. Drinfeld, A new realization of Yangians and quantum affine algebras, Soviet Math. Dokl. 36 (1988), No. 2, 212–216.
Z. N. C. Ha, Exact dynamical correlation functions of Calogero-Suth-erland model and one-dimensional fractional statistics, Phys. Rev. Lett. 73 (1994), No. 13, 1574–1577.
Z. N. C. Ha, Fractional statistics in one dimension: view from an exactly solvable model, Nucl. Phys. B 435 (1995), No. 3, 604–636.
M. Jimbo and T. Miwa, Solitons and infinite-dimensional Lie algebras, Pubi. Res. Inst. Math. Sci. 19 (1983), No. 3, 943–1001.
V. G. Kac and A. K. Raina, Bombay Lectures on Highest Weight Representations of Infinite-Dimensional Lie Algebras, Adv. Ser. Math. Phys., Vol. 2, World Sci. Publishing Co., Inc.ÉE; Teaneck, NJ, 1987.
Y. Kato, Green function of the Sutherland model with SU(2) internal symmetry, preprint.
F. Lesage, V. Pasquier, and D. Serban, Dynamical correlation functions in the Calogero-Sutherland model, Nucl. Phys. B 435 (1995), No. 3, 585–603.
I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd ed., Oxford Math. Monogr., Oxford Univ. Press, New York, 1995.
I. G. Macdonald, Affine Hecke algebras and orthogonal polynomials, Séminaire Bourbaki. Vol. 1994/95, Astérisque, Vol. 237, Soc. Math. France, Paris, 1996, pp. 189–207.
M. Nazarov and V. Tarasov, Representations of Yangians with Gelfand-Zetlin bases, J. Reine Angew. Math. 496 (1998), 181–212.
K. Takemura and D. Uglov, The orthogonal eigenbasis and norms of eigenvectors in the spin Calogero-Sutherland model, J. Phys. A 1997(30), No. 10, 3685–3717, solv-int/9611006.
D. Uglov, Yangian Gelfand-Zetlin bases, gl N-Jack polynomials and computation of dynamical correlation functions in the spin Calogero-Sutherland model, Commun. Math. Phys. 191 (1998), No. 3, 663–696.
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Uglov, D. (2000). Yangian Gelfand—Zetlin Bases, gl N -Jack Polynomials, and Computation of Dynamical Correlation Functions in the Spin Calogero—Sutherland Model. In: van Diejen, J.F., Vinet, L. (eds) Calogero—Moser— Sutherland Models. CRM Series in Mathematical Physics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1206-5_31
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DOI: https://doi.org/10.1007/978-1-4612-1206-5_31
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