Proceedings of the 28-th International Conference on Formal Power Series and Algebraic Combinatorics
4-8 Jul 2016 Vancouver, British Columbia (Canada)
Matrix product and sum rule for Macdonald polynomials
Luigi Cantini  1  , Jan De Gier  2  , Michael Wheeler  2  
1 : Laboratoire de Physique Théorique et Modélisation  (LPTM)  -  Website
Université de Cergy Pontoise, Centre National de la Recherche Scientifique
Université de Cergy-Pontoise 2 avenue Adolphe Chauvin, Pontoise 95302 Cergy-Pontoise cedex -  France
2 : Department of Mathematics and Statistics [Melbourne]  -  Website
The University of Melbourne Parkville, VIC, 3010 -  Australie

We present a new, explicit sum formula for symmetric Macdonald polynomials Pλ and show that they can be written as a trace over a product of (infinite dimensional) matrices. These matrices satisfy the Zamolodchikov– Faddeev (ZF) algebra. We construct solutions of the ZF algebra from a rank-reduced version of the Yang–Baxter algebra. As a corollary, we find that the normalization of the stationary measure of the multi-species asymmetric exclusion process is a Macdonald polynomial with all variables set equal to one. 

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