2015/05/31 by Luigi Cantini, Jan de Gier, Michael Wheeler · 1 citation
Physics and Astronomy · Mathematics · #math-ph #math.CO #math.MP #math.RT
paper · pdf · doi:10.1088/1751-8113/48/38/384001
published as J. Phys. A: Math. Theor. 48 (2015) 384001 · 27 pages; typos corrected, references added and some better conventions adopted in v2
arxiv created 2015/07/27 · arxiv updated 2015/09/30
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.