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A bijective proof of a factorization formula for Macdonald polynomials at roots of unity

Abstract : 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.
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F. Descouens, H. Morita, Y. Numata. A bijective proof of a factorization formula for Macdonald polynomials at roots of unity. 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008), 2008, Viña del Mar, Chile. pp.471-482. ⟨hal-01185126⟩

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