A Littlewood-Richardson rule for Macdonald polynomials

TL;DR

This paper develops a combinatorial Littlewood-Richardson rule for Macdonald polynomials using alcove walk combinatorics, providing a new product formula that generalizes several classical symmetric functions.

Contribution

It introduces a novel combinatorial approach to compute products of Macdonald polynomials, extending Littlewood-Richardson rules to the Macdonald setting.

Findings

Derived a product formula for Macdonald polynomials of general type.

Connected alcove walk combinatorics with Macdonald polynomial multiplication.

Unified various classical symmetric functions under a common framework.

Abstract

Macdonald polynomials are orthogonal polynomials associated to root systems, and in the type A case, the symmetric kind is a common generalization of Schur functions, Macdonald spherical functions, and Jack polynomials. We use the combinatorics of alcove walks to calculate products of monomials and intertwining operators of the double affine Hecke algebra. From this, we obtain a product formula for Macdonald polynomials of general type.