Nonsymmetric Macdonald polynomials via integrable vertex models

2019/04/15 by Borodin, Alexei, Wheeler, Michael · 1 citation
#Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1904.06804

Abstract

Starting from an integrable rank-n vertex model, we construct an explicit family of partition functions indexed by compositions μ= (μ₁,…,μ_n). Using the Yang-Baxter algebra of the model and a certain rotation operation that acts on our partition functions, we show that they are eigenfunctions of the Cherednik-Dunkl operators Y_i for all 1 ≤ i ≤ n, and are thus equal to nonsymmetric Macdonald polynomials E_μ. Our partition functions have the combinatorial interpretation of ensembles of coloured lattice paths which traverse a cylinder. Applying a simple bijection to such path ensembles, we show how to recover the well-known combinatorial formula for E_μ due to Haglund-Haiman-Loehr.

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