Matrix product formula for Macdonald polynomials

Abstract

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 are generalised probabilities for particle configurations of the multi-species asymmetric exclusion process, and 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.

Inhomogeneous discrete-time exclusion processes

“…Thus, discrete time inhomogeneous two species exclusion processes can be constructed and studied following the lines presented here. The generalization to N -species also seems possible [11].…”

A Summation Formula for Macdonald Polynomials

“…In this section we sketch the proof of (3), citing results from our earlier paper [1], where we obtained a matrix product formula for a family of non-symmetric polynomials f λ (x 1 , . .…”

“…As was demonstrated in [1], the symmetric Macdonald polynomial P λ is obtained as a sum over all polynomials f µ , whose composition is a distinct permutation of the partition λ:…”

“…We now recall the matrix product expression for f λ as obtained in [1]. Assume λ ⊆ r n and introduce the following family of (r − s + 2) × (r − s + 1) operator-valued matrices L (s) (x), 1 s r. We index rows by i ∈ {0, s, .…”

“…S λ = S n /S λ n is thus the set of all permutations which have a distinct action on λ. For example, for λ = (2, 2, 0, 0) we find (1,3,2,4), (1,3,4,2), (3, 1, 2, 4), (3, 1, 4, 2), (3, 4, 1, 2)} as the complete set of permutations with a distinguishable action on λ.…”