Journal of Lie Theory

no. 4
Vertices of Intersection Polytopes and Rays of Generalized Kostka Cones
Journal of Lie Theory, Volume 31 (2021) no. 4, pp. 1055-1070
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.
DOI: 10.5802/jolt.1211
Classification: 22E46, 05E10, 52A40
Keywords: Representation theory, convex geometry, Lie combinatorics, Kostka numbers, weight polytopes 
@article{JOLT_2021_31_4_a11,
     author = {M. Besson and S. Jeralds and J. Kiers},
     title = {Vertices of {Intersection} {Polytopes} and {Rays} of {Generalized} {Kostka} {Cones}},
     journal = {Journal of Lie Theory},
     pages = {1055--1070},
     year = {2021},
     volume = {31},
     number = {4},
     doi = {10.5802/jolt.1211},
     zbl = {1493.22012},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.1211/}
}
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M. Besson; S. Jeralds; J. Kiers. Vertices of Intersection Polytopes and Rays of Generalized Kostka Cones. Journal of Lie Theory, Volume 31 (2021) no. 4, pp. 1055-1070. doi: 10.5802/jolt.1211

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