Coxeter arrangements in three dimensions
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arXiv
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| Format: | Preprint |
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2015
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| _version_ | 1866917485153878016 |
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| author | Ehrenborg, Richard Klivans, Caroline Reading, Nathan |
| author_facet | Ehrenborg, Richard Klivans, Caroline Reading, Nathan |
| contents | Let ${\mathcal A}$ be a finite real linear hyperplane arrangement in three dimensions. Suppose further that all the regions of ${\mathcal A}$ are isometric. We prove that ${\mathcal A}$ is necessarily a Coxeter arrangement. As it is well known that the regions of a Coxeter arrangement are isometric, this characterizes three-dimensional Coxeter arrangements precisely as those arrangements with isometric regions. It is an open question whether this suffices to characterize Coxeter arrangements in higher dimensions. We also present the three families of affine arrangements in the plane which are not reflection arrangements, but in which all the regions are isometric. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1501_05991 |
| institution | arXiv |
| publishDate | 2015 |
| record_format | arxiv |
| spellingShingle | Coxeter arrangements in three dimensions Ehrenborg, Richard Klivans, Caroline Reading, Nathan Combinatorics 20F55 (Primary) 51F15, 52Cxx, 97G60 (Secondary) Let ${\mathcal A}$ be a finite real linear hyperplane arrangement in three dimensions. Suppose further that all the regions of ${\mathcal A}$ are isometric. We prove that ${\mathcal A}$ is necessarily a Coxeter arrangement. As it is well known that the regions of a Coxeter arrangement are isometric, this characterizes three-dimensional Coxeter arrangements precisely as those arrangements with isometric regions. It is an open question whether this suffices to characterize Coxeter arrangements in higher dimensions. We also present the three families of affine arrangements in the plane which are not reflection arrangements, but in which all the regions are isometric. |
| title | Coxeter arrangements in three dimensions |
| topic | Combinatorics 20F55 (Primary) 51F15, 52Cxx, 97G60 (Secondary) |
| url | https://arxiv.org/abs/1501.05991 |