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Vertices of Intersection Polytopes and Rays of Generalized Kostka Cones

Let $\mathscr{K}(G)$ be the rational cone generated by pairs $(λ, μ)$ where $λ$ and $μ$ are dominant integral weights and $μ$ is a nontrivial weight space in the representation $V_λ$ of $G$. We produce all extremal rays of $\mathscr{K}(G)$ by considering the vertices of corresponding intersection polytopes $IP_λ$, the set of points in $\mathscr{K}(G)$ with first coordinate $λ$. We show that vertices of $IP_{\varpi_i}$ arise as lifts of vertices coming from cones $\mathscr{K}(L)$ associated to simple Levi subgroups possessing the simple root $α_i$. 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.