Crystal energy functions via the charge in types A and C

TL;DR

This paper proves that the charge statistic for types A and C Macdonald polynomials equals the negative of the energy function on affine crystals, simplifying computation and providing new insights into crystal energy functions.

Contribution

It establishes the equality between charge and energy functions for types A and C, offering a more efficient way to compute crystal energies.

Findings

Charge equals negative energy function for types A and C

Simpler algorithm for computing charge

More efficient than recursive energy computation

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 prove that the charge is equal to the (negative of the) energy function on affine crystals. The algorithm for computing charge is much simpler and can be more efficiently computed than the recursive definition of energy in terms of the combinatorial R-matrix.