Topic
Normal polytope
About: Normal polytope is a research topic. Over the lifetime, 13 publications have been published within this topic receiving 177 citations.
Papers
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TL;DR: In this article, the authors introduce the notion of a favourable module for a complex unipotent algebraic group, whose properties are governed by the combinatorics of an associated polytope.
Abstract: We introduce the notion of a favourable module for a complex unipotent algebraic group, whose properties are governed by the combinatorics of an associated polytope. We describe two filtrations of the module, one given by the total degree on the PBW basis of the corresponding Lie algebra, the other by fixing a homogeneous monomial order on the PBW basis.
In the favourable case a basis of the module is parameterized by the lattice points of a normal polytope. The filtrations induce flat degenerations of the corresponding flag variety to its abelianized version and to a toric variety, the special fibres of the degenerations being projectively normal and arithmetically Cohen-Macaulay. The polytope itself can be recovered as a Newton-Okounkov body. We conclude the paper by giving classes of examples for favourable modules.
66 citations
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TL;DR: These faces are normal polytopes and their Minkowski sum is compatible with tensor products, which implies that flat degenerations of the corresponding Schubert varieties to PBW degenerated and toric varieties are obtained.
Abstract: We study certain faces of the normal polytope introduced by Feigin, Littelmann and the author whose lattice points parametrize a monomial basis of the PBW-degenerated of simple modules for $\mathfrak{sl}_{n+1}$. We show that lattice points in these faces parametrize monomial bases of PBW-degenerated Demazure modules associated to Weyl group elements satisfying a certain closure property, for example Kempf elements. These faces are again normal polytopes and their Minkowski sum is compatible with tensor products, which implies that we obtain flat degenerations of the corresponding Schubert varieties to PBW degenerated and toric varieties.
21 citations
TL;DR: In this paper, it was shown that every normal polytope is unimodularly equivalent to a face of a normal Gorenstein Fano polytopes of dimension d+1.
Abstract: It is known that every integral convex polytope is unimodularly equivalent to a face of some Gorenstein Fano polytope. It is then reasonable to ask whether every normal polytope is unimodularly equivalent to a face of some normal Gorenstein Fano polytope. In the present paper, it is shown that, by giving new classes of normal Gorenstein Fano polytopes, each order polytope as well as each chain polytope of dimension d is unimodularly equivalent to a facet of some normal Gorenstein Fano polytopes of dimension d+1. Furthermore, investigation on combinatorial properties, especially, Ehrhart polynomials and volume of these new polytopes will be achieved. Finally, some curious examples of Gorenstein Fano polytopes will be discovered.
18 citations
TL;DR: In this article, it was shown that every normal polytope is unimodularly equivalent to a face of a normal Gorenstein Fano polytopes of dimension n + 1.
Abstract: It is known that every integral convex polytope is unimodularly equivalent to a face of some Gorenstein Fano polytope. It is then reasonable to ask whether every normal polytope is unimodularly equivalent to a face of some normal Gorenstein Fano polytope. In the present paper, it is shown that, by giving new classes of normal Gorenstein Fano polytopes, each order polytope as well as each chain polytope of dimension $d$ is unimodularly equivalent to a facet of some normal Gorenstein Fano polytopes of dimension $d + 1$. Furthermore, investigation on combinatorial properties, especially, Ehrhart polynomials and volume of these new polytopes will be achieved. Finally, some curious examples of Gorenstein Fano polytopes will be discovered.
16 citations
TL;DR: In this article, the authors studied the PBW filtration on irreducible finite-dimensional representations for the Lie algebra of type B n and showed that there exists a normal polytope such that the lattice points of this polytoope parametrize a basis of the corresponding associated graded space.
14 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2019 | 1 |
| 2018 | 1 |
| 2017 | 3 |
| 2016 | 3 |
| 2015 | 2 |
| 2014 | 1 |