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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
86/2020

The universality of the resonance arrangement and its Betti numbers

Lukas Kühne

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.

Received:
Aug 25, 2020
Published:
Aug 28, 2020
MSC Codes:
05B35, 52B40, 14N20, 52C35
Keywords:
matroids, resonance arrangement, all-subsets arrangement, maximal unbalanced families, Betti numbers

Related publications

inJournal
2021 Journal Open Access
Lukas Kühne

The universality of the resonance arrangement and its Betti numbers

In: Séminaire lotharingien de combinatoire, 85B (2021), p. 75