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An intersection matrix for affine hyperplane arrangements
Algebraic Combinatorics, Volume 9 (2026) no. 2, pp. 499-512

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–Webster’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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.474
Classification: 05B35, 52C35, 52C40
Keywords: hyperplane arrangements, matroids, Schechtman-Varchenko, determinant formula
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
Eberhardt, Jens Niklas; Mautner, Carl. An intersection matrix for affine hyperplane arrangements. Algebraic Combinatorics, Volume 9 (2026) no. 2, pp. 499-512. doi: 10.5802/alco.474
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