Skip to main content

A New Generalisation of Macdonald Polynomials

Communications in Mathematical Physics volume 352, pages 773–804 (2017)Cite this article

Abstract

We introduce a new family of symmetric multivariate polynomials, whose coefficients are meromorphic functions of two parameters (q, t) and polynomial in a further two parameters (u, v). We evaluate these polynomials explicitly as a matrix product. At u = v = 0 they reduce to Macdonald polynomials, while at q = 0, u = v = s they recover a family of inhomogeneous symmetric functions originally introduced by Borodin.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

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

References

  1. Arita C., Ayyer A., Mallick K., Prolhac S.: Generalized matrix Ansatz in the multispecies exclusion process—partially asymmetric case. J. Phys. A Math. Theor. 45, 195001 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Bazhanov V.V., Kashaev R.M., Mangazeev V.V., Stroganov Yu.G.: \({(Z_N\times)^{n-1}}\) generalization of the chiral Potts model. Commun. Math. Phys. 138(2), 393–408 (1991)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Borodin, A.: Private communication.

  4. Borodin, A.: On a family of symmetric rational functions. arXiv:1410.0976

  5. Borodin A., Corwin I.: Macdonald processes. Prob. Theory Relat. Fields 158, 225–400 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Borodin, A., Petrov, L.: Higher spin six vertex model and symmetric rational functions. arXiv:1601.05770

  7. Bosnjak G., Mangazeev V.V.: Construction of R-matrices for symmetric tensor representations related to \({U_q(\widehat{sl_n})}\). J. Phys. A Math. Theor. 49, 495204 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Corwin I., Petrov L.: Stochastic higher spin vertex models on the line. Commun. Math. Phys. 343(2), 651–700 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Crampe, N., Ragoucy, E., Vanicat, M.: Integrable approach to simple exclusion processes with boundaries. Review and progress. J. Stat. Mech. P11032 (2014). arXiv:1408.5357

  13. Drinfeld, V.G.: Quantum groups. In: Proceedings of ICM-86 (Berkeley, USA), vol. 1. American Mathematical Society, Providence, pp. 798–820 (1987)

  14. Drinfeld, V.G.: Quasi-Hopf Algebra. Algebra and Analysis. Peterbg. Math. J. 1(6), 1419–1457 (1990)

  15. Faddeev, L.D.: Quantum completely integrable models in field theory. In: Problems of Quantum Field Theory, R2-12462, Dubna, pp. 249–299 (1979)

  16. Faddeev, L.D.: Quantum completely integrable models in field theory. In: Contemporary Mathematical Physics, vol. IC, pp. 107–155 (1980)

  17. Foda O., Wheeler M.: Colour-independent partition functions in coloured vertex models. Nucl. Phys. B 871, 330–361 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. Haiman M.: Hilbert schemes, polygraphs and the Macdonald positivity conjecture. J. Am. Math. Soc. 14, 941–1006 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hayashi T.: Q-analogues of Clifford and Weyl algebras-spinor and oscillator representations of quantum enveloping algebras. Commun. Math. Phys. 127, 129–144 (1990)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. Inoue, R., Kuniba, A., Okado, M.: A quantization of box-ball systems. Rev. Math. Phys. 16, 1227–1258 (2004). arXiv:nlin/0404047

  21. Jimbo M.: A q-analogue of \({U(gl(N+1))}\), Hecke algebra, and the Yang–Baxter equation. Lett. Math. Phys. 11, 247–252 (1986)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Jimbo M.: Quantum R-matrix for the generalized Toda system. Commun. Math. Phys. 102, 537–547 (1986)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Kasatani, M., Takeyama, Y.: The quantum Knizhnik–Zamolodchikov equation and non-symmetric Macdonald polynomials. Funkcialaj ekvacioj. Ser. Internacia 50, 491–509 (2007). arXiv:math/0608773

  24. Korff C.: Cylindric versions of specialised Macdonald polynomials and a deformed Verlinde algebra. Commun. Math. Phys. 318, 173–246 (2013)

    Article  ADS  MATH  Google Scholar 

  25. Kulish P.P., Mudrov A.I.: On twisting solutions to the Yang–Baxter equation. Czechoslov. J. Phys. 50(1), 115–122 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. Kuniba A., Mangazeev V.V., Maruyama S., Okado M.: Stochastic R matrix for \({U_q(A^{(1)}_n)}\). Nucl. Phys. B 913, 248–277 (2016)

    Article  ADS  MATH  Google Scholar 

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

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

    MATH  Google Scholar 

  29. Mangazeev V.: On the Yang–Baxter equation for the six-vertex model. Nucl. Phys. B 882, 70–96 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Mangazeev V.: Q-operators in the six-vertex model. Nucl. Phys. B 886, 166–184 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  32. Palev T.D.: A Holstein–Primakoff and a Dyson realization for the quantum algebra \({U_q(sl(n+1))}\). J. Phys. A Math. Gen. 31(22), 5145 (1998)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. Povolotsky, A.M.: On the integrability of zero-range chipping models with factorized steady states, J. Phys. A Math. Theor. 46, 465205 (2013)

  34. Prolhac, S., Evans, M.R., Mallick, K.: Matrix product solution of the multispecies partially asymmetric exclusion process. J. Phys. A Math. Theor. 42, 165004 (2009). arXiv:0812.3293

  35. Reshetikhin Yu N.: Multiparameter quantum groups and twisted quasitriangular Hopf algebras. Lett. Math. Phys 20, 331–335 (1990)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  36. Sasamoto T., Wadati M.: Exact results for one-dimensional totally asymmetric diffusion models. J. Phys. A Math. Gen. 31, 6057–6071 (1998)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  37. Takeyama, Y.: A deformation of affine Hecke algebra and integrable stochastic particle system. J. Phys. A Math. Theor. 47, 465203 (2014)

  38. Takeyama, Y.: Algebraic construction of multi-species q-Boson system. arXiv:1507.02033

  39. Tsuboi, Z.: Asymptotic representations and q-oscillator solutions of the graded Yang–Baxter equation related to Baxter Q-operators. Nucl. Phys. B 886, 1–30 (2014). arXiv:1205.1471

  40. Wheeler M., Zinn-Justin P.: Refined Cauchy/Littlewood identities and six-vertex model partition functions: III. Deformed bosons. Adv. Math. 299, 543–600 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  41. Zamolodchikov A.B., Zamolodchikov AI.B.: Two-dimensional factorizable S-matrices as exact solutions of some quantum field theory models. Ann. Phys. 120, 253–291 (1979)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. ARC Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS), School of Mathematics and Statistics, University of Melbourne, Parkville, VIC, 3010, Australia

    Alexandr Garbali, Jan de Gier & Michael Wheeler

Authors
  1. Alexandr Garbali
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jan de Gier
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Michael Wheeler
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michael Wheeler.

Additional information

Communicated by A. Borodin

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Garbali, A., de Gier, J. & Wheeler, M. A New Generalisation of Macdonald Polynomials. Commun. Math. Phys. 352, 773–804 (2017). https://doi.org/10.1007/s00220-016-2818-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-016-2818-1