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Journal of Lie Theory 31 (2021), No. 4, 1055--1070
Copyright Heldermann Verlag 2021

Vertices of Intersection Polytopes and Rays of Generalized Kostka Cones

Marc Besson
University of North Carolina, Chapel Hill, NC 27599, U.S.A.

Sam Jeralds
University of North Carolina, Chapel Hill, NC 27599, U.S.A.

Joshua Kiers
Ohio State University, Columbus, OH 43210, U.S.A.


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 a semisimple group $G$. We produce all extremal rays of $\mathcal{K}(G)$ by considering the vertices of corresponding intersection polytopes {\it IP}$_{\lambda}$, the set of points in $\mathcal{K}(G)$ with first coordinate $\lambda$. We show that vertices of {\it IP}$_{\varpi_i}$ 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.

Keywords: Representation theory, convex geometry, Lie combinatorics, Kostka numbers, weight polytopes.

MSC: 22E46, 05E10, 52A40.

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