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arXiv · 1505.00287

Matrix product formula for Macdonald polynomials

Luigi Cantini·Jan de Gier·Michael Wheeler
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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 $q=1$.

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Luigi Cantini, Jan de Gier, Michael Wheeler. 2015-07-27. Matrix product formula for Macdonald polynomials. https://doi.org/10.1088/1751-8113%2F48%2F38%2F384001

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