Orbit Structures of Weight Polytopes
Let $\Lambda$ be a dominant weight and let $ P(\Lambda)$ be the saturated weight set with highest weight $\Lambda$. The weight polytope associated with $\Lambda$ is defined as the convex hull of $ P(\Lambda)$. This paper provides an elementary proof that there is a natural bijection between the set of orbits of a weight polytope under action of the Weyl group and the set of parabolic subgroups of the Weyl group. The bijection is called the Putcha-Renner recipe of linear algebraic monoid theory.