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Communications in Mathematical Physics

, Volume 352, Issue 2, pp 773–804 | Cite as

A New Generalisation of Macdonald Polynomials

  • Alexandr Garbali
  • Jan de Gier
  • Michael WheelerEmail author
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.

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References

  1. 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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Borodin, A.: Private communication.Google Scholar
  4. 4.
    Borodin, A.: On a family of symmetric rational functions. arXiv:1410.0976
  5. 5.
    Borodin A., Corwin I.: Macdonald processes. Prob. Theory Relat. Fields 158, 225–400 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Borodin, A., Petrov, L.: Higher spin six vertex model and symmetric rational functions. arXiv:1601.05770
  7. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 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. 9.
    Cherednik I.: Double affine Hecke algebras and Macdonald’s conjectures. Ann. Math. 141, 191–216 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cherednik I.: Nonsymmetric Macdonald polynomials. Int. Math. Res. Not. 10, 483–515 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Corwin I., Petrov L.: Stochastic higher spin vertex models on the line. Commun. Math. Phys. 343(2), 651–700 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 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. 13.
    Drinfeld, V.G.: Quantum groups. In: Proceedings of ICM-86 (Berkeley, USA), vol. 1. American Mathematical Society, Providence, pp. 798–820 (1987)Google Scholar
  14. 14.
    Drinfeld, V.G.: Quasi-Hopf Algebra. Algebra and Analysis. Peterbg. Math. J. 1(6), 1419–1457 (1990)Google Scholar
  15. 15.
    Faddeev, L.D.: Quantum completely integrable models in field theory. In: Problems of Quantum Field Theory, R2-12462, Dubna, pp. 249–299 (1979)Google Scholar
  16. 16.
    Faddeev, L.D.: Quantum completely integrable models in field theory. In: Contemporary Mathematical Physics, vol. IC, pp. 107–155 (1980)Google Scholar
  17. 17.
    Foda O., Wheeler M.: Colour-independent partition functions in coloured vertex models. Nucl. Phys. B 871, 330–361 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Haiman M.: Hilbert schemes, polygraphs and the Macdonald positivity conjecture. J. Am. Math. Soc. 14, 941–1006 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Inoue, R., Kuniba, A., Okado, M.: A quantization of box-ball systems. Rev. Math. Phys. 16, 1227–1258 (2004). arXiv:nlin/0404047
  21. 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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Jimbo M.: Quantum R-matrix for the generalized Toda system. Commun. Math. Phys. 102, 537–547 (1986)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 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. 24.
    Korff C.: Cylindric versions of specialised Macdonald polynomials and a deformed Verlinde algebra. Commun. Math. Phys. 318, 173–246 (2013)ADSCrossRefzbMATHGoogle Scholar
  25. 25.
    Kulish P.P., Mudrov A.I.: On twisting solutions to the Yang–Baxter equation. Czechoslov. J. Phys. 50(1), 115–122 (2000)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 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)ADSCrossRefzbMATHGoogle Scholar
  27. 27.
    Macdonald, I.: A new class of symmetric functions. Publ. I.R.M.A. Strasbourg, Actes \({20^{\rm e}}\) Séminaire Lotharingien 131–171 (1988)Google Scholar
  28. 28.
    Macdonald I.: Symmetric functions and Hall polynomials, 2nd edn. Clarendon Press, Oxford (1995)zbMATHGoogle Scholar
  29. 29.
    Mangazeev V.: On the Yang–Baxter equation for the six-vertex model. Nucl. Phys. B 882, 70–96 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Mangazeev V.: Q-operators in the six-vertex model. Nucl. Phys. B 886, 166–184 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Opdam E.: Harmonic analysis for certain representations of graded Hecke algebras. Acta Math. 175, 75–121 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Povolotsky, A.M.: On the integrability of zero-range chipping models with factorized steady states, J. Phys. A Math. Theor. 46, 465205 (2013)Google Scholar
  34. 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. 35.
    Reshetikhin Yu N.: Multiparameter quantum groups and twisted quasitriangular Hopf algebras. Lett. Math. Phys 20, 331–335 (1990)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Sasamoto T., Wadati M.: Exact results for one-dimensional totally asymmetric diffusion models. J. Phys. A Math. Gen. 31, 6057–6071 (1998)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Takeyama, Y.: A deformation of affine Hecke algebra and integrable stochastic particle system. J. Phys. A Math. Theor. 47, 465203 (2014)Google Scholar
  38. 38.
    Takeyama, Y.: Algebraic construction of multi-species q-Boson system. arXiv:1507.02033
  39. 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. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 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)ADSCrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.ARC Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS), School of Mathematics and StatisticsUniversity of MelbourneParkvilleAustralia

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