The Universality of the Resonance Arrangement and its Betti Numbers

TL;DR

This paper explores the resonance arrangement's properties, demonstrating its universality for hyperplane arrangements, and derives explicit formulas for its Betti numbers, linking them to Stirling numbers of the second kind.

Contribution

It proves that any rational hyperplane arrangement is a minor of a resonance arrangement and provides exact formulas for its first two Betti numbers.

Findings

Resonance arrangements are universal for rational hyperplane arrangements.

Betti numbers of resonance arrangements are expressed via Stirling numbers of the second kind.

Exact formulas for the first two non-trivial Betti numbers are derived.

Abstract

The resonance arrangement $\mathcal{A}_n$ is the arrangement of hyperplanes which has all non-zero $0/1$-vectors in $\mathbb{R}^n$ as normal vectors. It is the adjoint of the Braid arrangement and is also called the all-subsets arrangement. The first result of this article shows that any rational hyperplane arrangement is the minor of some large enough resonance arrangement. Its chambers appear as regions of polynomiality in algebraic geometry, as generalized retarded functions in mathematical physics and as maximal unbalanced families that have applications in economics. One way to compute the number of chambers of any real arrangement is through the coefficients of its characteristic polynomial which are called Betti numbers. We show that the Betti numbers of the resonance arrangement are determined by a fixed combination of Stirling numbers of the second kind. Lastly, we develop exact formulas for the first two non-trivial Betti numbers of the resonance arrangement.