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Let $\lambda$ be a dominant weight of a finite dimensional simple Lie algebraand $W$ the Weyl group. The convex hull of $W\lambda$ is defined as the weightpolytope of $\lambda$. We provide a new proof that there is a natural bijectionbetween the set of orbits of the nonempty faces of the weight polytope underthe action of the Weyl group and the set of the connected subdiagrams of theextended Dynkin diagram that contain the extended node $\{-\lambda\}$. We showthat 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 ofa parabolic subgroup of $W$ acting on the dominant weight. In addition, we findthat the linear space spanned by a face is in fact spanned by roots.

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