Skip to main content
SpringerLink
Log in

A Summation Formula for Macdonald Polynomials

  • Published:
Letters in Mathematical Physics Aims and scope Submit manuscript

Abstract

We derive an explicit sum formula for symmetric Macdonald polynomials. Our expression contains multiple sums over the symmetric group and uses the action of Hecke generators on the ring of polynomials. In the special cases \({t = 1}\) and \({q = 0}\), we recover known expressions for the monomial symmetric and Hall–Littlewood polynomials, respectively. Other specializations of our formula give new expressions for the Jack and q–Whittaker polynomials.

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. Cantini, L., de Gier, J., Wheeler, M.: Matrix product formula for Macdonald polynomials. J. Phys. A: Math. Theor. 48, 384001 (2015). arXiv:1505.00287

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Cherednik I.: Nonsymmetric Macdonald polynomials. Int. Math. Res. Not. 10, 483–515 (1995)

    Article  MathSciNet  Google Scholar 

  4. Duchamp, G., Krob, D., Lascoux, A., Leclerc, B., Scharf, T., Thibon, J.Y.: Euler–Poincaré characteristic and polynomial representations of Iwahori–Hecke algebras. Publ. RIMS 31, 179–201 (1995)

  5. Haglund, J., Haiman, M., Loehr, N.: A combinatorial formula for Macdonald polynomials. J. Am. Math. Soc 18, 735–761 (2005). arXiv:math/0409538

  6. Haglund, J., Haiman, M., Loehr, N.: A combinatorial formula for non-symmetric Macdonald polynomials. Am. J. Math. 130, 359–383 (2008). arXiv:math/0601693

  7. Hivert F.: Hecke algebras, difference operators and quasi-symmetric functions. Adv. Math. 155, 181–238 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  8. Kirillov, A.N., Noumi, M.: q-Difference raising operators for Macdonald polynomials and the integrality of transition coefficients. In: Algebraic Methods and q-Special Functions, CRM Proceedings and Lecture Notes, vol. 22 (1999). arXiv:q-alg/9605005

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

  10. Macdonald, I.: A new class of symmetric functions. Publ. I.R.M.A. Strasbourg, Actes 20\({^\textrm{e}}\) Séminaire Lotharingien, vol. 131–171 (1988)

  11. Macdonald, I.: Symmetric functions and Hall polynomials, 2nd edn. Clarendon Press, Oxford (1995)

  12. Opdam E.: Harmonic analysis for certain representations of graded Hecke algebras. Acta Math. 175, 75–121 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ram, A., Yip, M.: A combinatorial formula for Macdonald polynomials. Adv. Math. 226, 309–31 (2011) arXiv:0803.1146

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, The University of Melbourne, Parkville, VIC, 3010, Australia

    Jan de Gier & Michael Wheeler

Authors
  1. Jan de Gier
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michael Wheeler
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michael Wheeler.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

de Gier, J., Wheeler, M. A Summation Formula for Macdonald Polynomials. Lett Math Phys 106, 381–394 (2016). https://doi.org/10.1007/s11005-016-0820-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11005-016-0820-3

Mathematics Subject Classification

Keywords

Search

Navigation