Demazure Formulas for Weight Polytopes

TL;DR

This paper explores the connection between Demazure character formulas and lattice polytope sums for simple Lie algebras, extending known formulas and investigating their analogs in polytope decompositions.

Contribution

It introduces Demazure operator-based expressions for lattice sums of weight polytopes in rank-2 and rank-3 simple Lie algebras, linking character formulas to polytope sums.

Findings

Derived Demazure operator expressions for rank-2 Lie algebra weight polytope sums

Extended the approach to the rank-3 algebra A_3

Identified similarities between Brion and Weyl character formulas

Abstract

The characters of simple Lie algebras are naturally decomposed into lattice polytope sums. The Brion formula for those polytope sums is remarkably similar to the Weyl character formula. Here we start to investigate if other character formulas have analogs for lattice polytope sums, by focusing on the Demazure character formulas. Using Demazure operators, we write expressions for the lattice sums of the weight polytopes of rank-2 simple Lie algebras, and the rank-3 algebra $A_3$.