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Modified Macdonald Polynomials and Integrability

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Abstract

We derive combinatorial formulae for the modified Macdonald polynomial \(H_{\lambda }(x;q,t)\) using coloured paths on a square lattice with quasi-cylindrical boundary conditions. The derivation is based on an integrable model associated to the quantum group of \(U_{q}(\widehat{sl_{n+1})}\).

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Notes

  1. Our definition differs from the definition of [25] by the inversion of t and the factor \(t^{n(\lambda )}\).

  2. The symmetry in \(x_1,\ldots ,x_N\) follows from the RLL equation satisfied by the weights (59) and the weights of the fused R-matrix (not given in this paper).

  3. This boundary condition does not apply to the paths of colour i in the column i.

  4. In order to match the data given by the partitions \(\nu _{i,j}\) and the lattice paths recall (98) and the relation between the indices \(\sigma \) and \(\nu \).

  5. In Sect. 4 we denoted this matrix \({\mathcal {L}}^{\lambda ,\mu }_{\lambda ',\mu '}(x,z)\).

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Acknowledgements

We gratefully acknowledge support from the Australian Research Council Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS), and MW acknowledges support by an Australian Research Council DECRA. We thank the program Non-equilibrium systems and special functions held at the MATRIX institute in Creswick, where part of this work was completed. We would like to thank Alexei Borodin, Jan de Gier, Atsuo Kuniba, Vladimir Mangazeev, Ole Warnaar and Paul Zinn-Justin for useful discussions.

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  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 & Michael Wheeler

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  1. Alexandr Garbali
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Correspondence to Alexandr Garbali.

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Communicated by A. Borodin.

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Garbali, A., Wheeler, M. Modified Macdonald Polynomials and Integrability. Commun. Math. Phys. 374, 1809–1876 (2020). https://doi.org/10.1007/s00220-020-03680-w

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