Abstract
We present a new, explicit sum formula for symmetric Macdonald polynomials Pλ 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.
Highlights
Symmetric Macdonald polynomials [16, 17] are a family of multivariable orthogonal polynomials indexed by partitions, whose coefficients depend rationally on two parameters q and t
Let mλ denote the monomial symmetric polynomial indexed by a partition λ, i.e. the symmetric polynomial defined as the sum of all monomials xμ = xμ11 · · · xμnn where μ ranges over all distinct permutations of λ = (λ1, . . . , λn)
Corollary 1 It follows from Lemma 1 that the symmetric Macdonald polynomial Pλ can be expressed as a sum over matrix product formulas
Summary
Symmetric Macdonald polynomials [16, 17] are a family of multivariable orthogonal polynomials indexed by partitions, whose coefficients depend rationally on two parameters q and t. The Macdonald polynomials are defined as follows: Definition 1 Let ·, · denote the Macdonald inner product on power sum symmetric functions ([17], Chapter VI, Equation (1.5)), where < denotes the dominance order on partitions ([17], Chapter I, Section 1). Macdonald polynomials can alternatively be defined as the unique eigenfunctions of certain linear difference operators acting on the space of all symmetric polynomials [17]. They can be expressed combinatorially as multivariable generating functions [8, 9, 21], or via symmetrization of non-symmetric Macdonald polynomials that are computed from Yang–Baxter graphs [14, 15]. In view of the relations for the generators, we can define Tσ unambiguously as any product of simple transpositions Ti which gives the permutation σ
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