From Macdonald polynomials to a charge s

The charge is an intricate statistic on words, introduced by Lascoux and Schützenberger, which provides positive combinatorial formulas for Lusztig’s q-analogue of weight multiplicities and the energy function on affine crystals of type A. Aiming to express these concepts for all Lie types, this paper presents a methodology to generalize the charge statistic in classical Lie types, focusing on types A and C. The method utilizes the recent Ram-Yip formula for Macdonald polynomials and the quantum Bruhat order within the associated Weyl group. In detailed analysis, the classical charge is recovered in type A, while a new statistic is introduced in type C. Through these developments, the author addresses the long-standing problem of extending the definition of a charge statistic beyond type A, thus contributing to the broader understanding of combinatorial structures associated with Macdonald polynomials. This study not only provides insight into the combinatorial and algebraic aspects of these polynomials but also opens new avenues for research in the intersection of representation theory and combinatorics.