Oriented Matroid Circuit Polytopes
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912174530625536 |
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| author | Escobar, Laura McWhirter, Jodi |
| author_facet | Escobar, Laura McWhirter, Jodi |
| contents | Matroids give rise to several natural constructions of polytopes. Inspired by this, we examine polytopes that arise from the signed circuits of an oriented matroid. We give the dimensions of these polytopes arising from graphical oriented matroids and their duals. Moreover, we consider polytopes constructed from cocircuits of oriented matroids generated by the positive roots in any type A root system. We give an explicit description of their face structure and determine the Ehrhart series. We also study an action of the symmetric group on these polytopes, giving a full description the subpolytopes fixed by each permutation. These type A polytopes are graphic zonotopes, are polar duals of symmetric edge polytopes, and also make an appearance in Stapledon's paper introducing Equivariant Ehrhart Theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_00108 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Oriented Matroid Circuit Polytopes Escobar, Laura McWhirter, Jodi Combinatorics 05A15, 05B35, 05C20, 11P21, 20F55, 52B20, 52C40 Matroids give rise to several natural constructions of polytopes. Inspired by this, we examine polytopes that arise from the signed circuits of an oriented matroid. We give the dimensions of these polytopes arising from graphical oriented matroids and their duals. Moreover, we consider polytopes constructed from cocircuits of oriented matroids generated by the positive roots in any type A root system. We give an explicit description of their face structure and determine the Ehrhart series. We also study an action of the symmetric group on these polytopes, giving a full description the subpolytopes fixed by each permutation. These type A polytopes are graphic zonotopes, are polar duals of symmetric edge polytopes, and also make an appearance in Stapledon's paper introducing Equivariant Ehrhart Theory. |
| title | Oriented Matroid Circuit Polytopes |
| topic | Combinatorics 05A15, 05B35, 05C20, 11P21, 20F55, 52B20, 52C40 |
| url | https://arxiv.org/abs/2501.00108 |