Vertices of Intersection Polytopes and Rays of Generalized Kostka Cones

Marc Besson¹ ; Sam Jeralds¹ ; Joshua Kiers²

¹University of North Carolina, Chapel Hill, NC 27599, U.S.A.
²Ohio State University, Columbus, OH 43210, U.S.A.

Journal of Lie Theory, Tome 31 (2021) no. 4, pp. 1055-1070

Résumé

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.

Détail

Classification : 22E46, 05E10, 52A40
Mots-clés : Representation theory, convex geometry, Lie combinatorics, Kostka numbers, weight polytopes