Strongly dominant weight polytopes are cubes

Abstract

For any root system of rank $r$, we study the "dominant weight polytope" $P^{\hat{I}}»$ associated with a strongly dominant weight $\hat{I}»$. We prove that $P^{\hat{I}}»$ is combinatorially equivalent to the $r$-dimensional cube. As an application, we give a new proof of the known formulas for Betti numbers of Peterson varieties in classical Lie types.

13 pages, 4 figures. Generalization of the main result included. Application to Peterson varieties included. Several small corrections