  We derive a sharp decomposition formula for the state polytope of the Hilbert
point and the Hilbert-Mumford index of reducible varieties by using the
decomposition of characters and basic convex geometry. This proof captures the
essence of the decomposition of the state polytopes in general, and
considerably simplifies an earlier proof by the author and Jaekwang Kim which
uses a careful analysis of initial ideals of reducible varieties.
