Set-valued tableaux for Macdonald polynomials

TL;DR

This paper develops a set-valued tableaux formula for Macdonald polynomials by translating the alcove walk formula, aiming to deepen the connection between Macdonald polynomial calculus and Schubert calculus.

Contribution

It introduces a new set-valued tableaux formula for Macdonald polynomials derived from alcove walk formulas, enhancing the combinatorial understanding.

Findings

Set-valued tableaux formula for Macdonald polynomials established

Strengthens the analogy between Macdonald polynomial calculus and Schubert calculus

Provides a combinatorial tool for future research in algebraic combinatorics

Abstract

Set-valued tableaux formulas play an important role in Schubert calculus. Using the box greedy reduced word for the construction of the Macdonald polynomials, we convert the alcove walk formula for Macdonald polynomials to a set-valued tableaux formula for Macdonald polynomials. Our hope is that providing set-valued tableaux formulas for Macdonald polynomials will help to strengthen the analogies and possible connections between the calculus of Macdonald polynomials and Schubert calculus.