Skip to main content
SpringerLink
Log in

Nonsymmetric interpolation macdonald polynomials and \( \mathfrak{g}\mathfrak{l}_n \) basic hypergeometric series

  • Published:
Transformation Groups Aims and scope Submit manuscript

Abstract

The Knop-Sahi interpolation Macdonald polynomials are inhomogeneous and nonsymmetric generalisations of the well-known Macdonald polynomials. In this paper we apply the interpolation Macdonald polynomials to study a new type of basic hypergeometric series of type \( \mathfrak{g}\mathfrak{l}_n \). Our main results include a new q-binomial theorem, a new q-Gauss sum, and several transformation formulae for \( \mathfrak{g}\mathfrak{l}_n \) series.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. G. E. Andrews, The Theory of Partitions, Encyclopedia of Mathematics and its Applications, Vol. 2, Addison-Wesley, Reading, MA, 1976. Russian transl.: Г. Эндрюс, Теорuя разбuенuй, Наука, М., 1982.

  2. T. H. Baker, P. J. Forrester, Transformation formulas for multivariable basic hyper- geometric series, Methods Appl. Anal. 6 (1999), 147–164.

    MATH  MathSciNet  Google Scholar 

  3. T. H. Baker, P. J. Forrester, Isomorphisms of type A affine Hecke algebras and multivariable orthogonal polynomials, Pacific J. Math. 194 (2000), 19–41.

    Article  MATH  MathSciNet  Google Scholar 

  4. I. Cherednik, A unification of Knizhnik-Zamolodchikov and Dunkl operators via affine Hecke algebras, Invent. Math. 106 (1991), 411–431.

    Article  MATH  MathSciNet  Google Scholar 

  5. I. Cherednik, Nonsymmetric Macdonald polynomials, Int. Math. Res. Not. (1995), no. 10, 483–515.

  6. G. Gasper, M. Rahman, Basic Hypergeometric Series, Encyclopedia of Mathematics and its Applications, Vol. 35, 2nd ed., Cambridge University Press, Cambridge, 2004.

    MATH  Google Scholar 

  7. R. A. Gustafson, Multilateral summation theorems for ordinary and basic hypergeometric series in U(n), SIAM J. Math. Anal. 18 (1987), 1576–1596.

    Article  MATH  MathSciNet  Google Scholar 

  8. J. Kaneko, q-Selberg integrals and Macdonald polynomials, Ann. Sci. École Norm. Sup. (4) 29 (1996), 583–637.

    MATH  Google Scholar 

  9. F. Knop, Integrality of two variable Kostka functions, J. Reine Angew. Math. 482 (1997), 177–189.

    MATH  MathSciNet  Google Scholar 

  10. F. Knop, Symmetric and nonsymmetric quantum Capelli polynomials, Comment. Math. Helv. 72 (1997), 84–100.

    MATH  MathSciNet  Google Scholar 

  11. A. Lascoux, Yang-Baxter graphs, Jack and Macdonald polynomials, Ann. Comb. 5 (2001), 397–424.

    Article  MATH  MathSciNet  Google Scholar 

  12. A. Lascoux, Symmetric Functions and Combinatorial Operators on Polynomials, CBMS Regional Conference Series in Mathematics, Vol. 99, Amer. Math. Soc., Providence, RI, 2003.

    MATH  Google Scholar 

  13. A. Lascoux, Macdonald and Schubert for dummies, available at www.ms.unimelb. edu.au/~degier/GandI08.php.

  14. A. Lascoux, M.-P. Schützenberger, Symmetry and flag manifolds, in: Invariant Theory, Lecture Notes in Mathematics, Vol. 996, Springer-Verlag, Berlin, 1983, pp. 118–144.

    Google Scholar 

  15. M. Lassalle, Coefficients binomiaux généralisés et polynômes de Macdonald, J. Funct. Anal. 158 (1998), 289–324.

    Article  MATH  MathSciNet  Google Scholar 

  16. G. Lusztig, Affine Hecke algebras and their graded version, J. Amer. Math. Soc. 2 (1989), 599–635.

    Article  MATH  MathSciNet  Google Scholar 

  17. I. G. Macdonald, A new class of symmetric functions, Sém. Lothar. Combin. B20a (1988), 41 pp.

    Google Scholar 

  18. I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd ed., Oxford University Press, New York, 1995.

    MATH  Google Scholar 

  19. I. G. Macdonald, Hypergeometric series II, unpublished manuscript.

  20. S. C. Milne, A q-analog of the 5 F 4(1) summation theorem for hypergeometric series well-poised in SU(n), Adv. in Math. 57 (1985), 14–33.

    Article  MATH  MathSciNet  Google Scholar 

  21. S. C. Milne, An elementary proof of the Macdonald identities for A (1) l , Adv. in Math. 57 (1985), 34–70.

    Article  MATH  MathSciNet  Google Scholar 

  22. S. C. Milne, Transformations of U(n + 1) multiple basic hypergeometric series, in: Physics and Combinatorics 1999, World Scientific, River Edge, NJ, 2001, pp. 201–243.

    Google Scholar 

  23. A. Okounkov, Binomial formula for Macdonald polynomials and applications, Math. Res. Lett. 4 (1997), 533–553.

    MATH  MathSciNet  Google Scholar 

  24. A. Okounkov, On Newton interpolation of symmetric functions: A characterization of interpolation Macdonald polynomials, Adv. Appl. Math. 20 (1998), 395–428.

    Article  MATH  MathSciNet  Google Scholar 

  25. A. Okounkov, BC-type interpolation Macdonald polynomials and binomial formula for Koornwinder polynomials, Transform. Groups 3 (1998), 181–207.

    Article  MATH  MathSciNet  Google Scholar 

  26. E. M. Rains, BC n -symmetric polynomials, Transform. Groups 10 (2005), 63–132.

    Article  MATH  MathSciNet  Google Scholar 

  27. E. M. Rains, BC n -symmetric Abelian functions, Duke Math. J. 135 (2006), 99–180.

    Article  MATH  MathSciNet  Google Scholar 

  28. E. M. Rains, Elliptic Littlewood identities, available at arXiv:0806.0871.

  29. S. Sahi, Interpolation, integrality, and a generalisation of Macdonald's polynomials, Int. Math. Res. Not. (1996), no. 10, 457–471.

  30. S. Sahi, A new scalar product for the nonsymmetric Jack polynomials, Int. Math. Res. Not. (1996), no. 20, 997–1004.

  31. S. Sahi, The binomial formula for nonsymmetric Macdonald polynomials, Duke Math. J. 94 (1998), 465–477.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. CNRS Institut Gaspard Monge, Université Paris-Est, Marne-La-Vallée, France

    Alain Lascoux

  2. Department of Mathematics, California Institute of Technology, Pasadena, CA, 91125, USA

    Eric M. Rains

  3. School of Mathematics and Physics, The University of Queensland, Brisbane, QLD 4072, Australia

    S. Ole Warnaar

Authors
  1. Alain Lascoux
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Eric M. Rains
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. Ole Warnaar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alain Lascoux.

Additional information

*Supported by the ANR project MARS (BLAN06-2 134516).

**Supported by the NSF grant DMS-0401387.

***Supported by the Australian Research Council.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lascoux, A., Rains, E.M. & Warnaar, S.O. Nonsymmetric interpolation macdonald polynomials and \( \mathfrak{g}\mathfrak{l}_n \) basic hypergeometric series. Transformation Groups 14, 613–647 (2009). https://doi.org/10.1007/s00031-009-9061-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00031-009-9061-1

Keywords

Search

Navigation