An intersection matrix for affine hyperplane arrangements

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 𝒪 (or more generally Kowalenko–Mautner’s category 𝒪 for oriented matroid programs). This paper is part of a broader program to categorify matroidal Schur algebras defined by Braden–Mautner.