An Intersection Matrix for Affine Hyperplane Arrangements

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.