Polytope sums and Lie characters

TL;DR

This paper introduces a novel approach using polytope theory to analyze Lie characters, revealing degeneracies in weight-multiplicities through polytope expansions and the Brion formula.

Contribution

It applies polytope sums to Lie theory, providing new insights into weight-multiplicities and their degeneracies beyond Weyl symmetry.

Findings

Polytope sums can be used to express Lie characters.

The Brion formula helps clarify degeneracies in weight-multiplicities.

Polytope expansions offer a transparent view of Lie algebra representations.

Abstract

A new application of polytope theory to Lie theory is presented. Exponential sums of convex lattice polytopes are applied to the characters of irreducible representations of simple Lie algebras. The Brion formula is used to write a polytope expansion of a Lie character, that makes more transparent certain degeneracies of weight-multiplicities beyond those explained by Weyl symmetry.