Colorful combinatorics and Macdonald polynomials

TL;DR

This paper reveals that Macdonald polynomials can be understood as generating functions of colored words with a cocharge statistic, connecting combinatorics, representation theory, and algebraic geometry.

Contribution

It introduces a new combinatorial framework linking Macdonald polynomials to colored words and extends classical concepts like cocharge and plactic monoid to broader contexts.

Findings

Macdonald polynomials are generating functions of colored words with cocharge.

Extended the plactic monoid to include coloring and related it to crystal theory.

Applied the framework to Garsia-Haiman modules and K-theoretic Schubert calculus.

Abstract

The non-negative integer cocharge statistic on words was introduced in the 1970's by Lascoux and Sch\"utzenberger to combinatorially characterize the Hall-Littlewood polynomials. Cocharge has since been used to explain phenomena ranging from the graded decomposition of Garsia-Procesi modules to the cohomology structure of the Grassman variety. Although its application to contemporary variations of these problems had been deemed intractable, we prove that the two-parameter, symmetric Macdonald polynomials are generating functions of a distinguished family of colored words. Cocharge adorns one parameter and the second measures its deviation from cocharge on words without color. We use the same framework to expand the plactic monoid, apply Kashiwara's crystal theory to various Garsia-Haiman modules, and to address problems in K-theoretic Schubert calculus.