Polytope sums and Lie characters
  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. 
M. A. Walton