Report
Vertices of Intersection Polytopes and Rays of Generalized Kostka Cones
| Title: | Vertices of Intersection Polytopes and Rays of Generalized Kostka Cones |
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| Authors: | Besson, Marc, Jeralds, Sam, Kiers, Joshua |
| Publication Year: | 2020 |
| Collection: | Mathematics |
| Subject Terms: | Mathematics - Representation Theory, 22E46 (Primary) 05E10, 52A40 (Secondary) |
| Description: | Let $\mathscr{K}(G)$ 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 $\mathscr{K}(G)$ by considering the vertices of corresponding intersection polytopes $IP_{\lambda}$, the set of points in $\mathscr{K}(G)$ with first coordinate $\lambda$. 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 $\alpha_i$. 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. Comment: 14 pages, with 10 figures; v2 includes a new averaging formula which simplifies several proofs |
| Document Type: | Working Paper |
| DOI: | 10.48550/arXiv.2004.09711 |
| Access URL: | http://arxiv.org/abs/2004.09711 |
| Accession Number: | edsarx.2004.09711 |
| Database: | arXiv |
| DOI: | 10.48550/arXiv.2004.09711 |
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