The universality of the resonance arrangement and its Betti numbers
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Submission date: 25. Aug. 2020
MSC-Numbers: 05B35, 52B40, 14N20, 52C35
Keywords and phrases: matroids, resonance arrangement,, all-subsets arrangement, maximal unbalanced families, Betti numbers
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The resonance arrangement 𝒜n is the arrangement of hyperplanes which has all non-zero 0∕1-vectors in ℝn 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.