  The weight systems of finite-dimensional representations of complex, simple
Lie algebras exhibit patterns beyond Weyl-group symmetry. These patterns occur
because weight systems can be decomposed into lattice polytopes in a natural
way. Since lattice polytopes are relatively simple, this decomposition is
useful, in addition to being more economical than the decomposition into single
weights. An expansion of characters into polytope sums follows from the
polytope decomposition of weight systems. We study this polytope expansion
here. A new, general formula is given for the polytope sums involved. The
combinatorics of the polytope expansion are analyzed; we point out that they
are reduced from those of the Weyl character formula (described by the Kostant
partition function) in an optimal way. We also show that the weight
multiplicities can be found easily from the polytope multiplicities, indicating
explicitly the equivalence of the two descriptions. Finally, we demonstrate the
utility of the polytope expansion by showing how polytope multiplicities can be
used in the calculation of tensor product decompositions, and subalgebra
branching rules.
