From Macdonald Polynomials to a Charge Statistic beyond Type A

TL;DR

This paper develops a new approach to generalize the charge statistic, originally specific to type A, to other classical Lie types using Macdonald polynomials and the Ram-Yip formula, leading to new combinatorial insights.

Contribution

It introduces a method to extend the charge statistic beyond type A, specifically applying it to type C, based on recent advances in Macdonald polynomial formulas and quantum Bruhat order.

Findings

Recovered classical charge in type A

Defined a new charge statistic in type C

Connected charge to Macdonald polynomials and quantum Bruhat order

Abstract

The charge is an intricate statistic on words, due to Lascoux and Schutzenberger, which gives positive combinatorial formulas for Lusztig's q-analogue of weight multiplicities and the energy function on affine crystals, both of type A. As these concepts are defined for all Lie types, it has been a long-standing problem to express them based on a generalization of charge. I present a method for addressing this problem in classical Lie types, based on the recent Ram-Yip formula for Macdonald polynomials and the quantum Bruhat order on the corresponding Weyl group. The details of the method are carried out in type A (where we recover the classical charge) and type C (where we define a new statistic).