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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.06008 |
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| _version_ | 1866910517932589056 |
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| author | Eberhardt, Jens Niklas Mautner, Carl |
| author_facet | Eberhardt, Jens Niklas Mautner, Carl |
| contents | For a real affine hyperplane arrangement, we define an integer intersection matrix with a natural $q$-deformation related to the intersections of bounded chambers of the arrangement. By connecting the integer matrix to a bilinear form of Schechtman-Varchenko, we show that there is a closed formula for its determinant that only depends on the combinatorics of the underlying matroid. We conjecture an analogous formula for its $q$-deformation. Our work also applies more generally in the setting of affine oriented matroids.
Additionally, we give a representation-theoretic interpretation of our $q$-intersection matrix using Braden-Licata-Proudfoot-Websters's hypertoric category $\mathcal{O}$ (or more generally Kowalenko-Mautner's category $\mathcal{O}$ for oriented matroid programs). This paper is part of a broader program to categorify matroidal Schur algebras defined by Braden-Mautner. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_06008 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An Intersection Matrix for Affine Hyperplane Arrangements Eberhardt, Jens Niklas Mautner, Carl Combinatorics Representation Theory 52C40, 52C35 For a real affine hyperplane arrangement, we define an integer intersection matrix with a natural $q$-deformation related to the intersections of bounded chambers of the arrangement. By connecting the integer matrix to a bilinear form of Schechtman-Varchenko, we show that there is a closed formula for its determinant that only depends on the combinatorics of the underlying matroid. We conjecture an analogous formula for its $q$-deformation. Our work also applies more generally in the setting of affine oriented matroids. Additionally, we give a representation-theoretic interpretation of our $q$-intersection matrix using Braden-Licata-Proudfoot-Websters's hypertoric category $\mathcal{O}$ (or more generally Kowalenko-Mautner's category $\mathcal{O}$ for oriented matroid programs). This paper is part of a broader program to categorify matroidal Schur algebras defined by Braden-Mautner. |
| title | An Intersection Matrix for Affine Hyperplane Arrangements |
| topic | Combinatorics Representation Theory 52C40, 52C35 |
| url | https://arxiv.org/abs/2407.06008 |