Crystal energy via charge

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.

Keywords

• affine crystals
• energy function
• charge
• kashiwara-nakashima tableaux
• macdonald polynomials
• [info.info-dm] computer science [cs]/discrete mathematics [cs.dm]