Affine Demazure Weight Polytopes and Twisted Bruhat Orders
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914740789313536 |
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| author | Besson, Marc Jeralds, Sam Kiers, Joshua |
| author_facet | Besson, Marc Jeralds, Sam Kiers, Joshua |
| contents | For an untwisted affine Kac-Moody Lie algebra $\mathfrak{g}$ with Cartan and Borel subalgebras $\mathfrak{h} \subset \mathfrak{b} \subset \mathfrak{g}$, affine Demazure modules are certain $U(\mathfrak{b})$-submodules of the irreducible highest-weight representations of $\mathfrak{g}$. We introduce here the associated affine Demazure weight polytopes, given by the convex hull of the $\mathfrak{h}$-weights of such a module. Using methods of geometric invariant theory, we determine inequalities which define these polytopes; these inequalities come in three distinct flavors, specified by the standard, opposite, or semi-infinite Bruhat orders. We also give a combinatorial characterization of the vertices of these polytopes lying on an arbitrary face, utilizing the more general class of twisted Bruhat orders. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_03142 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Affine Demazure Weight Polytopes and Twisted Bruhat Orders Besson, Marc Jeralds, Sam Kiers, Joshua Representation Theory Algebraic Geometry Combinatorics 14M15, 22E66, 17B67, 05E10, 52A40 For an untwisted affine Kac-Moody Lie algebra $\mathfrak{g}$ with Cartan and Borel subalgebras $\mathfrak{h} \subset \mathfrak{b} \subset \mathfrak{g}$, affine Demazure modules are certain $U(\mathfrak{b})$-submodules of the irreducible highest-weight representations of $\mathfrak{g}$. We introduce here the associated affine Demazure weight polytopes, given by the convex hull of the $\mathfrak{h}$-weights of such a module. Using methods of geometric invariant theory, we determine inequalities which define these polytopes; these inequalities come in three distinct flavors, specified by the standard, opposite, or semi-infinite Bruhat orders. We also give a combinatorial characterization of the vertices of these polytopes lying on an arbitrary face, utilizing the more general class of twisted Bruhat orders. |
| title | Affine Demazure Weight Polytopes and Twisted Bruhat Orders |
| topic | Representation Theory Algebraic Geometry Combinatorics 14M15, 22E66, 17B67, 05E10, 52A40 |
| url | https://arxiv.org/abs/2404.03142 |