Demazure crystals, Kirillov-Reshetikhin crystals, and the energy function

TL;DR

This paper explores the relationship between Demazure crystals and Kirillov-Reshetikhin crystals, revealing how an energy function encodes Demazure characters and connects to Macdonald polynomials and q-Whittaker functions.

Contribution

It demonstrates that the isomorphism between Demazure crystals and tensor products of Kirillov-Reshetikhin crystals preserves the affine grading via an energy function, providing new character formulas.

Findings

Energy function encodes Demazure characters

Isomorphism preserves affine grading

Applications to Macdonald polynomials

Abstract

It has previously been shown that, at least for non-exceptional Kac-Moody Lie algebras, there is a close connection between Demazure crystals and tensor products of Kirillov-Reshetikhin crystals. In particular, certain Demazure crystals are isomorphic as classical crystals to tensor products of Kirillov-Reshetikhin crystals via a canonically chosen isomorphism. Here we show that this isomorphism intertwines the natural affine grading on Demazure crystals with a combinatorially defined energy function. As a consequence, we obtain a formula of the Demazure character in terms of the energy function, which has applications to Macdonald polynomials and q-deformed Whittaker functions.