Demazure formula for $A_n$ Weyl polytope sums

TL;DR

This paper derives a new Demazure formula for Weyl polytope sums in type A_n Lie algebras, connecting representation characters, Brion formulas, and Demazure operators.

Contribution

It introduces a novel Demazure operator-based formula for Weyl polytope sums in $A_n$ Lie algebras, valid for all dominant weights and ranks.

Findings

The formula applies to all dominant integrable highest weights.

It holds for all ranks n in type A_n Lie algebras.

Establishes a new link between Weyl polytopes and Demazure operators.

Abstract

The weights of finite-dimensional representations of simple Lie algebras are naturally associated with Weyl polytopes. Representation characters decompose into multiplicity-free sums over the weights in Weyl polytopes. The Brion formula for these Weyl polytope sums is remarkably similar to the Weyl character formula. Moreover, the same Lie characters are also expressible as Demazure character formulas. This motivates a search for new expressions for Weyl polytope sums, and we prove such a formula involving Demazure operators. It applies to the Weyl polytope sums of the simple Lie algebras $A_n$, for all dominant integrable highest weights and all ranks $n$.