Cones of Hyperplane Arrangements

Abstract

Hyperplane arrangements dissect $\R^n$ into connected components called chambers, and a well-known theorem of Zaslavsky counts chambers as a sum of nonnegative integers calledWhitney numbers of the first kind. His theorem generalizes to count chambers within any cone defined as the intersection of a collection of halfspaces from the arrangement, leading to a notion of Whitney numbers for each cone. This thesis concerns cones of hyperplane arrangement in two ways. First we consider cones within the braid arrangement, consisting of the reflecting hyperplanes $x_i=x_j$ inside $\R^n$ for the symmetric group, thought of as the type $A_{n-1}$ reflection group. Here,\begin{itemize} \item cones correspond to posets, \item chambers within the cone correspond to linear extensions of the poset, \item the Whitney numbers of the cone interestingly refine the number of linear extensions of the poset. \end{itemize} We interpret this refinement explicitly for two families of posets: width two posets, and disjoint unions of chains. In the latter case, this gives a geometric re-interpretation to Foata's theory of cycle decomposition for multiset permutations, and leads to a simple generating function compiling these Whitney numbers. Secondly, we give an interpretation of the coefficients the Poincaré polynomial of a cone of an arbitrary arrangement via the Varchenko-Gel'fand ring, which is the ring of functions from the chambers of the arrangement to the integers with pointwise addition and multiplication. Varchenko and Gel'fand gave a simple presentation for this ring for an arbitrary arrangement, along with a filtration and associated graded ring whose Hilbert series is the Poincaré polynomial.We generalize these results to cones defined by intersections of halfspaces of some of the hyperplanes and prove a novel result for the Varchenko-Gel'fand ring of an arrangement: when the arrangement is supersolvable the associated graded ring of the arrangement is Koszul.