Colorful combinatorics and Macdonald polynomials

Abstract

The non-negative integer cocharge statistic on words was introduced in the 1970s by Lascoux and Schützenberger to combinatorially characterize the Hall–Littlewood polynomials. Cocharge has since been used to explain phenomena ranging from the graded decomposition of Garsia–Procesimodules to the cohomology structure of the Grassmann 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 measure 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.