Cross-section Lattices of ${\mathcal J}$-irreducible Monoids and Orbit Structures of Weight Polytopes

Let $\lambda$ be a dominant weight of a finite dimensional simple Lie algebra and $W$ the Weyl group. The convex hull of $W\lambda$ is defined as the weight polytope of $\lambda$. We provide a new proof that there is a natural bijection between the set of orbits of the nonempty faces of the weight polytope under the action of the Weyl group and the set of the connected subdiagrams of the extended Dynkin diagram that contain the extended node $\{-\lambda\}$. We show that each face of the polytope can be transformed to a standard parabolic face. We also show that a standard parabolic face is the convex hull of the orbit of a parabolic subgroup of $W$ acting on the dominant weight. In addition, we find that the linear space spanned by a face is in fact spanned by roots.