An Intersection Matrix for Affine Hyperplane Arrangements

TL;DR

This paper introduces an intersection matrix for affine hyperplane arrangements, explores its determinant formula based on matroid combinatorics, and connects it to representation theory and categorification of matroidal algebras.

Contribution

It defines a new integer intersection matrix with a q-deformation, derives a determinant formula, and links it to representation theory and categorification of matroidal structures.

Findings

Determinant formula depends only on matroid combinatorics

Established a q-deformation related to bounded chambers

Connected the matrix to hypertoric category O and categorification

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.