Note on the decomposition of states

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.