Combinatorics of $m=1$ Grasstopes
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866912357199904768 |
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| author | Mandelshtam, Yelena Pavlov, Dmitrii Pratt, Elizabeth |
| author_facet | Mandelshtam, Yelena Pavlov, Dmitrii Pratt, Elizabeth |
| contents | A Grasstope is the image of the totally nonnegative Grassmannian $\text{Gr}_{\geq 0}(k,n)$ under a linear map $\text{Gr}(k,n)\dashrightarrow \text{Gr}(k,k+m)$. This is a generalization of the amplituhedron, a geometric object of great importance to calculating scattering amplitudes in physics. The amplituhedron is a Grasstope arising from a totally positive linear map. While amplituhedra are relatively well-studied, much less is known about general Grasstopes. We study Grasstopes in the $m=1$ case and show that they can be characterized as unions of cells of a hyperplane arrangement satisfying a certain sign variation condition, extending work of Karp and Williams. Inspired by this characterization, we also suggest a notion of a Grasstope arising from an arbitrary oriented matroid. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2307_09603 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Combinatorics of $m=1$ Grasstopes Mandelshtam, Yelena Pavlov, Dmitrii Pratt, Elizabeth Combinatorics Algebraic Geometry 05E14, 14N10, 14M15 A Grasstope is the image of the totally nonnegative Grassmannian $\text{Gr}_{\geq 0}(k,n)$ under a linear map $\text{Gr}(k,n)\dashrightarrow \text{Gr}(k,k+m)$. This is a generalization of the amplituhedron, a geometric object of great importance to calculating scattering amplitudes in physics. The amplituhedron is a Grasstope arising from a totally positive linear map. While amplituhedra are relatively well-studied, much less is known about general Grasstopes. We study Grasstopes in the $m=1$ case and show that they can be characterized as unions of cells of a hyperplane arrangement satisfying a certain sign variation condition, extending work of Karp and Williams. Inspired by this characterization, we also suggest a notion of a Grasstope arising from an arbitrary oriented matroid. |
| title | Combinatorics of $m=1$ Grasstopes |
| topic | Combinatorics Algebraic Geometry 05E14, 14N10, 14M15 |
| url | https://arxiv.org/abs/2307.09603 |