Matrix product formula for Macdonald polynomials

TL;DR

This paper presents a matrix product formula for symmetric Macdonald polynomials, connecting algebraic solutions of deformed equations with combinatorial models and stochastic processes.

Contribution

It introduces a novel matrix product representation for Macdonald polynomials derived from solutions to deformed Knizhnik--Zamolodchikov equations.

Findings

Provides a basis of polynomial solutions indexed by compositions.

Establishes a connection between Macdonald polynomials and solvable lattice models.

Shows that stationary states of multi-species exclusion processes relate to Macdonald polynomials at q=1.

Abstract

We derive a matrix product formula for symmetric Macdonald polynomials. Our results are obtained by constructing polynomial solutions of deformed Knizhnik--Zamolodchikov equations, which arise by considering representations of the Zamolodchikov--Faddeev and Yang--Baxter algebras in terms of $t$-deformed bosonic operators. These solutions form a basis of the ring of polynomials in $n$ variables, whose elements are indexed by compositions. For weakly increasing compositions (anti-dominant weights), these basis elements coincide with non-symmetric Macdonald polynomials. Our formulas imply a natural combinatorial interpretation in terms of solvable lattice models. They also imply that normalisations of stationary states of multi-species exclusion processes are obtained as Macdonald polynomials at $q=1$.