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arXiv · 2407.06008

An Intersection Matrix for Affine Hyperplane Arrangements

Jens Niklas Eberhardt·Carl Mautner
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Abstract

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.

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Jens Niklas Eberhardt, Carl Mautner. 2024-07-08. An Intersection Matrix for Affine Hyperplane Arrangements. https://arxiv.org/abs/2407.06008

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