Hyperplane Arrangements and Algebra

Hyperplanes can be used to describe polytopes, (oriented) matroids, Coxeter groups, and other combinatorial objects. Collections of hyperplanes dissect the ambient vector space into connected components called regions, and Varchenko-Gelfand introduced a ring from the regions of a real arrangement by considering all maps from the set of regions to the integers, with pointwise addition and multiplication. Varchenko-Gelfand gave an interesting presentation for this ring as (a quotient of) a polynomial ring, and an interesting interpretation of this presentation is given by Moseley. In this talk, we consider the intersection of a hyperplane arrangement with an open, convex set and recover analogues both of Varchenko-Gelfand's presentation and of Moseley's topological interpretation.