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Deformed Macdonald-Ruijsenaars Operators and Super Macdonald Polynomials

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Abstract

It is shown that the deformed Macdonald-Ruijsenaars operators can be described as the restrictions on certain affine subvarieties of the usual Macdonald- Ruijsenaars operator in infinite number of variables. The ideals of these varieties are shown to be generated by the Macdonald polynomials related to Young diagrams with special geometry. The super Macdonald polynomials and their shifted version are introduced; the combinatorial formulas for them are given.

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References

  1. Sergeev A.N., Veselov A.P.: Deformed quantum Calogero-Moser problems and Lie superalgebras. Commun. Math. Phys. 245(2), 249–278 (2004)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  2. Sergeev A.N., Veselov A.P.: Generalised discriminants, deformed Calogero-Moser-Sutherland operators and super-Jack polynomials. Adv. Math. 192(2), 341–375 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  3. Ruijsenaars S.N.M.: Complete integrability of relativistic Calogero-Moser systems and elliptic function identities. Commun. Math. Phys. 110(2), 191–213 (1987)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  4. Macdonald I.: Symmetric Functions and Hall Polynomials 2nd edition. Oxford Univ. Press, Oxford (1995)

    Google Scholar 

  5. Knop F., Sahi S.: Difference equations and symmetric polynomials defined by their zeros. Internat. Math. Res. Notes 10, 473–486 (1996)

    Article  MathSciNet  Google Scholar 

  6. Knop F.: Symmetric and non-symmetric quantum Capelli polynomials. Comment. Math. Helv. 72(1), 84–100 (1997)

    MATH  MathSciNet  Google Scholar 

  7. Sahi S.: Interpolation, integrality, and a generalization of Macdonald’s polynomials. Internat. Math. Res. Notices 10, 457–471 (1996)

    Article  MathSciNet  Google Scholar 

  8. Okounkov A.: (Shifted) Macdonald polynomials : q-integral representation and combinatorial formula. Compositio Math. 112(2), 147–182 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  9. Feigin B., Jimbo M., Miwa T., Mukhin E.: Symmetric polynomials vanishing on the shifted diagonals and Macdonald polynomials. Int. Math. Res. Not. no. 18, 1015–1034 (2003)

    Article  MathSciNet  Google Scholar 

  10. Chalykh O.A.: Macdonald polynomials and algebraic integrability. Adv. Math. 166(2), 193–259 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  11. Veselov A.P., Feigin M.V., Chalykh O.A.: New integrable deformations of quantum Calogero - Moser problem. Russ. Math. Surv. 51(3), 185–186 (1996)

    Article  MathSciNet  Google Scholar 

  12. Cherednik I.: Double affine Hecke algebras and Macdonald’s conjectures. Ann. of Math. (2) 141(1), 191–216 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  13. Kirillov A.N., Noumi M.: Affine Hecke algebras and raising operators for Macdonald polynomials. Duke Math. J. 93(1), 1–39 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  14. Kasatani, M., Miwa, T., Sergeev, A.N., Veselov, A.P.: Coincident root loci and Jack and Macdonald polynomials for special values of the parameters. In: Jack, Hall-Littlewood and Macdonald Polynomials, 207–225, Contemp. Math., 417, Providence, RI: Amer. Math. Soc., 2006, pp. 207–225

  15. Atiyah M., Macdonald I.G.: Introduction to Commutative Algebra. Addison-Wesley, Reading, MA (1969)

    MATH  Google Scholar 

  16. Haglund J., Haiman M., Loehr N.: A combinatorial formula for Macdonald polynomials. J. Amer. Math. Soc. 18(3), 735–761 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  17. Ram, A., Yip, M.: A combinatorial formula for Macdonald polynomials. http://arXiv.org/abs/0803.1146v1[math-co.], 2008

  18. Sergeev, A.N., Veselov, A.P.: BC-infinity Calogero-Moser operator and super Jacobi polynomials. http://arXiv.org/abs/0807.3858v2[math-ph], 2008

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Authors and Affiliations

  1. Department of Mathematical Sciences, Loughborough University, Loughborough, LE11 3TU, UK

    A. N. Sergeev & A. P. Veselov

  2. Steklov Institute of Mathematics, Fontanka 27, St., Petersburg, 191023, Russia

    A. N. Sergeev

  3. Landau Institute for Theoretical Physics, Moscow, Russia

    A. P. Veselov

Authors
  1. A. N. Sergeev
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  2. A. P. Veselov
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Correspondence to A. P. Veselov.

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Communicated by L. Takhtajan

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Sergeev, A.N., Veselov, A.P. Deformed Macdonald-Ruijsenaars Operators and Super Macdonald Polynomials. Commun. Math. Phys. 288, 653–675 (2009). https://doi.org/10.1007/s00220-009-0779-3

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  • DOI: https://doi.org/10.1007/s00220-009-0779-3

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