A bijective proof of a factorization formula for Macdonald polynomials at roots of unity
F. Descouens (1) (2), H. Morita (3), Y. Numata (4)
(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]
Abstract
English
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.
French
Nous présentons une preuve combinatoire de la formule de factorisation des polynômes de Macdonald modifiés $\widetilde{H}_{\lambda} (X;q,t)$ quand $t$ est spécialisé à une racine primitive de l'unité. Notre preuve se restreint au cas particulier des partitions $\lambda$ n'ayant que deux colonnes. On utilise principalement l'interprétation combinatoire de Haglund, Haiman and Loehr donnant le développement de $\widetilde{H}_{\lambda} (X;q,t)$ sur la base des fonctions monomiales.
