Yangian Gelfand—Zetlin Bases, glN-Jack Polynomials, and Computation of Dynamical Correlation Functions in the Spin Calogero—Sutherland Model
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.
KeywordsSymmetric Polynomial Laurent Polynomial Macdonald Polynomial Jack Polynomial Density Correlation Function
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- 5.I. Cherednik, Nonsymmetric Macdonald polynomials, Internat. Math. Res. Notices (1995), No. 10, 483–515.Google Scholar
- 8.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.Google Scholar
- 13.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.Google Scholar
- 14.Y. Kato, Green function of the Sutherland model with SU(2) internal symmetry, preprint.Google Scholar
- 16.I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd ed., Oxford Math. Monogr., Oxford Univ. Press, New York, 1995.Google Scholar
- 17.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.Google Scholar
- 19.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.Google Scholar