Skip to main content
SpringerLink
Log in

Specializing Koornwinder polynomials to Macdonald polynomials of type B, C, D and BC

  • Published:
Journal of Algebraic Combinatorics Aims and scope Submit manuscript

Abstract

We study the specializations of parameters in Koornwinder polynomials to obtain Macdonald polynomials associated to the subsystems of the affine root system of type \((C_n^\vee ,C_n)\) in the sense of Macdonald (Affine Hecke algebras and orthogonal polynomials, Cambridge tracts in mathematics, Cambridge Univ Press, 2003), and summarize them in what we call the specialization table. As a verification of our argument, we check the specializations to type B, C and D via Ram–Yip type formulas of non-symmetric Koornwinder and Macdonald 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.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Askey, R., Wilson, J.: Some basic hypergeometric orthogonal polynomials that generalize the Jacobi polynomials, Mem. Amer. Math. Soc., 54(319) (1985)

  2. Cherednik, I.: Double affine Hecke algebras, Knizhnik-Zamolodchikov equations, and Macdonald’s operators. Int. Math. Res. Not. 9, 171–179 (1992)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Cherednik, I.: Non-symmetric Macdonald polynomials. Int. Math. Res. Not. 10, 483–515 (1995)

    Article  MATH  Google Scholar 

  5. Cherednik, I.: Macdonald’s evaluation conjectures and difference Fourier transform. Inv. Math. 122, 119–145 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cherednik, I.: Intertwining operators of double affine Hecke algebras. Sel. Math. New Ser. 3, 459–495 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cherednik, I.: Double affine Hecke algebras, London Math. Soc. Lecture Note Series, 319, Cambridge Univ. Press, Cambridge (2005)

  8. Chihara, M.: Demazure slices of type \(A^{(2)}_{2l}\). Algebr. Represent. Theory 25, 491–519 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  9. Feigin, B., Hoshino, A., Noumi, M., Shibahara, J., Shiraishi, J.: Tableau formulas for one-row Macdonald polynomials of types \(C_n\) and \(D_n\). SIGMA Symmetry Integrability Geom. Methods Appl. 11, 100 (2015)

    MATH  Google Scholar 

  10. Gasper, G., Rahman, M.: Basic hypergeometric series, 2nd. ed., Encyclopedia of Mathematics and its Applications 96, Cambridge University Press (2004)

  11. Haiman, M.: Cherednik algebras, Macdonald polynomials and combinatorics, International Congress of Mathematicians. Vol. III, 843–872, Eur. Math. Soc., Zürich, (2006)

  12. Hoshino, A., Shiraishi, J.: Macdonald polynomials of type Cn with one-column diagrams and deformed Catalan numbers. SIGMA Symmetry Integrability Geom. Methods Appl. 14, 33 (2018)

    MATH  Google Scholar 

  13. Hoshino, A., Shiraishi, J.: Branching rules for Koornwinder polynomials with one column diagrams and matrix inversions. SIGMA Symmetry Integrability Geom. Methods Appl. 16, 28 (2020)

    MathSciNet  MATH  Google Scholar 

  14. Ion, B.: Nonsymmetric Macdonald polynomials and Demazure characters. Duke Math. J. 116(2), 299–318 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kac, V.G.: Infinite-Dimensional Lie Algebras, 3rd edn. Cambridge University Press, Cambridge (1990)

    Book  MATH  Google Scholar 

  16. Koekoek, L.: Swarttouw: Hypergeometric Orthogonal Polynomials and Their \(q\)-Analogues. Springer Monographs in Mathematics. Springer, Berlin (2010)

    Book  Google Scholar 

  17. Koornwinder, T.H.: Askey-Wilson polynomials for root systems of type BC. Contemp. Math. 138, 189–204 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  18. Koornwinder, T.H.: Charting the Askey and \(q\)-Askey schemes, Lecture in AMS Special Session on The Legacy of Dick Askey, (6–9 January, 2021). The slide is available from https://staff.fnwi.uva.nl/t.h.koornwinder/art/sheets/

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

    Article  MathSciNet  MATH  Google Scholar 

  20. Macdonald, I.G.: Affine root systems and Dedekind’s \(\eta \)-function. Inv. Math. 15, 161–174 (1972)

    MathSciNet  Google Scholar 

  21. Macdonald, I.G.: Orthogonal polynomials associated with root systems, preprint, 1987; typed and published in the Séminaire Lotharingien de Combinatoire, 45, B45a (2000); available from arXiv:math/0011046

  22. Macdonald, I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. Oxford Univ Press (1995)

  23. Macdonald, I.G.: Affine Hecke Algebras and Orthogonal Polynomials, Cambridge Tracts in Mathematics, vol. 157. Cambridge Univ Press (2003)

    Book  Google Scholar 

  24. Noumi, M.: Macdonald-Koornwinder polynomials and affine Hecke rings (in Japanese), in Various Aspects of Hypergeometric Functions (Kyoto, 1994), RIMS Kôkyûroku 919, Research Institute for Mathematical Sciences, Kyoto Univ., Kyoto, 44–55 (1995)

  25. Noumi, M., Stokman, J.V.: Askey-Wilson polynomials: an affine Hecke algebraic approach. In: Alvarez-Nodarse, R., Marcellan, F., Van Assche, W. (eds.) Laredo Lectures on Orthogonal Polynomials and Special Functions, pp. 111–144. Nova Science Publishers (2004)

    MATH  Google Scholar 

  26. Orr, D., Shimozono, M.: Specializations of nonsymmetric Macdonald-Koornwinder polynomials. J. Algebraic Combin. 47(1), 91–127 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  27. Ram, A.: Alcove walks, Hecke algebras, spherical functions, crystals and column strict tableaux. Pure Appl. Math. Q. 2(4), 963–1013 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  28. Ram, A., Yip, M.: A combinatorial formula for Macdonald polynomials. Adv. Math. 226, 309–331 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  29. Rosengren, H.: Proofs of some partition identities conjectured by Kanade and Russell, Ramanujan J. (2021), published online. arXiv:1912.03689

  30. Sahi, S.: Nonsymmetric Koornwinder polynomials and duality. Ann. Math. 150, 267–282 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  31. Sahi, S.: Some properties of Koornwinder polynomials. Contemp. Math. 254, 395–411 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  32. Sanderson, Y.B.: On the Connection Between Macdonald Polynomials and Demazure Characters. J. Algebraic Combin. 11, 269–275 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  33. Stokman, J.V.: Koornwinder Polynomials and Affine Hecke Algebras. Int. Math. Res. Not. 19, 1005–1042 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  34. Stokman, J.V: Lecture Notes on Koornwinder polynomials, in Laredo Lectures on Orthogonal Polynomials and Special Functions, 145–207, Adv. Theory Spec. Funct. Orthogonal Polynomials, Nova Sci. Publ. Hauppauge, NY, (2004)

  35. Stokman, J.V.: Macdonald-Koornwinder polynomials, in Encyclopedia of Special Functions: The Askey-Bateman Project: Volume 2, Multivariable Special Functions, Cambridge University Press, (2020); provisional version available from arXiv:1111.6112

  36. van Diejen, J.: Self-dual Koornwinder-Macdonald polynomials. Invent. Math. 126(2), 319–339 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  37. Yamaguchi, K.: A Littlewood-Richardson rule for Koornwinder polynomials. J. Algebraic Combin. 56(2), 335–381 (2022). https://doi.org/10.1007/s10801-022-01114-5

Download references

Acknowledgements

The authors would like to express gratitude to Professor Masatoshi Noumi for valuable comments. In particular, Remarks 2.3.2 and 2.4.2 are added as a result of the correspondences with him. The authors would also like to thank the referees for valuable comments, suggestions and the mention to the reference [18], which enabled them to add the Sect. 2.6 of the rank one (Askey–Wilson) case. K.Y. is supported by JSPS Fellowships for Young Scientists (No. 22J11816), and S.Y. is supported by JSPS KAKENHI Grant Number 19K03399.

Author information

Authors and Affiliations

  1. Nagoya University, Nagoya, Japan

    Kohei Yamaguchi & Shintarou Yanagida

Authors
  1. Kohei Yamaguchi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shintarou Yanagida
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shintarou Yanagida.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yamaguchi, K., Yanagida, S. Specializing Koornwinder polynomials to Macdonald polynomials of type B, C, D and BC. J Algebr Comb 57, 171–226 (2023). https://doi.org/10.1007/s10801-022-01165-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10801-022-01165-8

Keywords

Search

Navigation