Abstract
The Ram–Yip formula for Macdonald polynomials (at t=0) provides a statistic which we call charge. In types ${A}$ and ${C}$ it can be defined on tensor products of Kashiwara–Nakashima single column crystals. In this paper we show that the charge is equal to the (negative of the) energy function on affine crystals. The algorithm for computing charge is much simpler than the recursive definition of energy in terms of the combinatorial ${R}$-matrix. La formule de Ram et Yip pour les polynômes de Macdonald (à t = 0) fournit une statistique que nous appelons la charge. Dans les types ${A}$ et ${C}$, elle peut être définie sur les produits tensoriels des cristaux pour les colonnes de Kashiwara–Nakashima. Dans ce papier, nous montrons que la charge est égale à (l'opposé de) la fonction d'énergie sur cristaux affines. L'algorithme pour calculer la charge est bien plus simple que la définition récursive de l'énergie en fonction de la ${R}$-matrice combinatoire.
Highlights
The energy function of affine crystals is an important grading used in one-dimensional configuration sums [7, 8] and generalized Kostka polynomials [31, 33, 34]
This leads us to the role of the charge statistic, which can be calculated very efficiently, as it only involves the detection of descents and the computation of arm lengths of cells in Young diagrams
The goal of this paper is to show in an efficient, conceptual way that the charge in [18] coincides with the energy function on the corresponding tensor products of KR crystals
Summary
The energy function of affine crystals is an important grading used in one-dimensional configuration sums [7, 8] and generalized Kostka polynomials [31, 33, 34]. A generalization of (2) to -laced types was given in [9]; in types A and D, this result is sharpened in [30, Section 9.2] by replacing the Kostka–Foulkes polynomials with the corresponding one-dimensional configuration sums (which are generating functions for the energy) Both (1) in type A and (2) are expressed combinatorially in terms of the Lascoux–Schutzenberger charge, whereas the type C charge given by (1) is a new statistic. To compare our work with the previous papers on charge and energy, let us first say that our results apply to arbitrary vertices in a tensor product of KN columns, not just to the highest weight elements (with respect to the nonzero arrows), that are used in the work involving Kostka–Foulkes polynomials. A long version of this paper containing all proofs has appeared [22]
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