Cones of Hyperplane Arrangements through the Varchenko-Gel’fand Ring

Abstract: The coefficients of the characteristic polynomial of an arrangement in a real vector space have many interpretations. An interesting one is provided by the Varchenko-Gel’fand ring, which is the ring of functions from the chambers of the arrangement to the integers with pointwise multiplication. Varchenko and Gel’fand gave a simple presentation for this ring, along with a filtration whose associated graded ring has its Hilbert function given by the coefficients of the characteristic polynomial. We generalize these results to cones defined by intersections of halfspaces of some of the hyperplanes. Time permitting, we will discuss Varchenko–Gel’fand analogues of some well-known results in the Orlik–Solomon algebra regarding Koszulity and supersolvable arrangements.

combinatorics

Audience: researchers in the topic

York University Applied Algebra Seminar