Geodesic
A combinatorial approach to Macdonald q, t-symmetry via the Carlitz bijection
Discrete mathematics & theoretical computer science, DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016), DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016) (2020).

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We investigate the combinatorics of the symmetry relation H μ(x; q, t) = H μ∗ (x; t, q) on the transformed Macdonald polynomials, from the point of view of the combinatorial formula of Haglund, Haiman, and Loehr in terms of the inv and maj statistics on Young diagram fillings. By generalizing the Carlitz bijection on permutations, we provide a purely combinatorial proof of the relation in the case of Hall-Littlewood polynomials (q = 0) for the coefficients of the square-free monomials in the variables x. Our work in this case relates the Macdonald inv and maj statistics to the monomial basis of the modules Rμ studied by Garsia and Procesi. We also provide a new proof for the full Macdonald relation in the case when μ is a hook shape. 
@article{DMTCS_2020_special_379_a1,
     author = {Gillespie, Maria Monks},
     title = {A combinatorial approach to {Macdonald} q, t-symmetry via the {Carlitz} bijection},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016)},
     year = {2020},
     doi = {10.46298/dmtcs.6319},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.6319/}
}
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Gillespie, Maria Monks. A combinatorial approach to Macdonald q, t-symmetry via the Carlitz bijection. Discrete mathematics & theoretical computer science, DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016), DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016) (2020). doi : 10.46298/dmtcs.6319. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.6319/

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