Vertices of Intersection Polytopes and Rays of Generalized Kostka Cones

Abstract

Let $\mathcal{K}(G)$ be the rational cone generated by pairs $(\lambda ,\mu )$ where $\lambda$ and $\mu$ are dominant integral weights and $\mu$ is a nontrivial weight space in the representation $V_{\lambda }$ of $G$. We produce all extremal rays of $\mathcal{K}(G)$ by considering the vertices of corresponding intersection polytopes $I{P}_{\lambda }$, the set of points in $\mathcal{K}(G)$ with first coordinate $\lambda$. We show that vertices of $I{P}_{{\varpi }_i}$ arise as lifts of vertices coming from cones $\mathcal{K}(L)$ associated to simple Levi subgroups possessing the simple root ${\alpha }_i$. As corollaries we obtain a complete description of all extremal rays, as well as polynomial formulas describing the numbers of extremal rays depending on type and rank.

14 pages, with 10 figures; v2 includes a new averaging formula which simplifies several proofs