Matrix product and sum rule for Macdonald polynomials

TL;DR

This paper introduces a new explicit sum formula for Macdonald polynomials, expressing them as traces over matrix products satisfying the ZF algebra, linking symmetric functions with integrable algebraic structures.

Contribution

It provides a novel sum formula for Macdonald polynomials and connects them to the ZF algebra and Yang--Baxter algebra, revealing new algebraic insights.

Findings

Macdonald polynomials can be expressed as traces over matrix products.

Matrices involved satisfy the Zamolodchikov--Faddeev algebra.

Normalization of stationary measure in multi-species ASEP is a Macdonald polynomial.

Abstract

We present a new, explicit sum formula for symmetric Macdonald polynomials $P_\lambda$ 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.