Root Cones and the Resonance Arrangement

We study the connection between triangulations of a type $A$ root polytope and resonance arrangement, hyperplane arrangement that shows up in surprising number contexts. Despite an elementary definition for chambers has only been computed to $n=8$ dimensional case. focus on data structures labeling chambers, such as sign vectors sets alternating trees, with aim at better understanding structure and, particular, enumerating its chambers. Along way, we make connections similar (and similarly difficult) enumeration questions. With viewpoint, relate polynomiality Kostant partition function. clarify threshold functions. In show base-2 logarithm is asymptotically $n^2$.