Preprint 86/2020

The universality of the resonance arrangement and its Betti numbers

Lukas Kühne

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Submission date: 25. Aug. 2020
Pages: 18
MSC-Numbers: 05B35, 52B40, 14N20, 52C35
Keywords and phrases: matroids, resonance arrangement,, all-subsets arrangement, maximal unbalanced families, Betti numbers
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Link to arXiv: See the arXiv entry of this preprint.

The resonance arrangement 𝒜n is the arrangement of hyperplanes which has all non-zero 01-vectors in 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.

03.09.2020, 02:19