Basis enumeration of hyperplane arrangements up to symmetries

Given a symmetry group acting on the hyperplanes of an arrangement, our goal is to report a single basis from each orbit of bases induced by this group. In this paper we extend previous techniques for finding the (feasible) bases of polyhedra up to symmetry, and for computing the symmetry groups of polyhedra, to the setting of hyperplane arrangements. We present some preliminary experiments with a C++ implementation of these techniques called Basil. These results show substantial speedups compared to a previous polyhedra only system using the computer algebra system GAP. We also measure the speedup due to a Gram matrix invariant, and show that the overhead of symmetry testing, while substantial, is dominated by the savings in reduced pivoting.