F. Descouens ; H. Morita ; Y. Numata

A bijective proof of a factorization formula for Macdonald polynomials at roots of unity
dmtcs:3593 
Discrete Mathematics & Theoretical Computer Science,
January 1, 2008,
DMTCS Proceedings vol. AJ, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008)

https://doi.org/10.46298/dmtcs.3593
A bijective proof of a factorization formula for Macdonald polynomials at roots of unityArticle
Authors: F. Descouens ^{1,}^{2}; H. Morita ^{3}; Y. Numata ^{4}
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F. Descouens;H. Morita;Y. Numata
1 Fields Institute for Research In Mathematical Sciences
2 Department of Mathematics and Statistics [Toronto]
3 Oyama National College of Technology
4 Department of Mathematics [Sapporo]
We give a combinatorial proof of the factorization formula of modified Macdonald polynomials $\widetilde{H}_{\lambda} (X;q,t)$ when $t$ is specialized at a primitive root of unity. Our proof is restricted to the special case where $\lambda$ is a two columns partition. We mainly use the combinatorial interpretation of Haiman, Haglund and Loehr giving the expansion of $\widetilde{H}_{\lambda} (X;q,t)$ on the monomial basis.
Nicholas A. Loehr;Elizabeth Niese, 2012, A Bijective Proof of a Factorization Formula for Specialized Macdonald Polynomials, Annals of combinatorics, 16, 4, pp. 815828, 10.1007/s0002601201625.