Root Cones and the Resonance Arrangement

Abstract

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

Computing characteristic polynomials of hyperplane arrangements with symmetries

“…A formula for their number of chambers remains elusive, let alone one for their Betti numbers. Nonetheless, partial formulas and bounds exist [4,17,30,45].…”

“…This count is derived from the computation of other important combinatorial invariants: Betti numbers and characteristic polynomials [1,19,28,36,42,44]. While most arrangements admit few combinatorial symmetries [38], most arrangements of interest do [17,39,45].…”