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The Universality of the Resonance Arrangement and its Betti Numbers

Lukas Kühne

TL;DR

This work studies the resonance arrangement A_n, the all-subsets hyperplane arrangement, and proves a universality principle: any rational hyperplane arrangement (equivalently any representable matroid over Q) appears as a minor of A_n for a suitable n. The authors connect chambers, Betti numbers, and combinatorial structures via the broken circuit complex, showing that for each fixed i the Betti numbers b_i(A_n) are linear combinations of Stirling numbers S(n+1,k) with coefficients c_{i,k} independent of n, and they provide explicit bounds on these coefficients. They derive exact formulas for the first two nontrivial Betti numbers, b_2(A_n) and b_3(A_n), in closed form using Stirling numbers and power sums, leveraging detailed analyses of circuits and broken circuits, including tetrahedron and rectangle types. The results yield bounds on R_n (the number of chambers) and illuminate the structure of A_n, with implications for algebraic geometry, mathematical physics, and economics through the interpretation of chambers and their polynomiality regions.

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.

The Universality of the Resonance Arrangement and its Betti Numbers

TL;DR

This work studies the resonance arrangement A_n, the all-subsets hyperplane arrangement, and proves a universality principle: any rational hyperplane arrangement (equivalently any representable matroid over Q) appears as a minor of A_n for a suitable n. The authors connect chambers, Betti numbers, and combinatorial structures via the broken circuit complex, showing that for each fixed i the Betti numbers b_i(A_n) are linear combinations of Stirling numbers S(n+1,k) with coefficients c_{i,k} independent of n, and they provide explicit bounds on these coefficients. They derive exact formulas for the first two nontrivial Betti numbers, b_2(A_n) and b_3(A_n), in closed form using Stirling numbers and power sums, leveraging detailed analyses of circuits and broken circuits, including tetrahedron and rectangle types. The results yield bounds on R_n (the number of chambers) and illuminate the structure of A_n, with implications for algebraic geometry, mathematical physics, and economics through the interpretation of chambers and their polynomiality regions.

Abstract

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

Paper Structure

This paper contains 9 sections, 14 theorems, 38 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Resonance Arrangement
  3. Chambers of $\mathcal{A}_n$
  4. Matroids and their Minors
  5. Universality of the Resonance Arrangement
  6. The Broken Circuit Complex
  7. Proof of \ref{['thm:betti_stirling']}
  8. The Betti Number $b_2(\mathcal{A}_n)$
  9. The Betti Number $b_3(\mathcal{A}_n)$

Key Result

Theorem 1.2

Let $\mathcal{B}$ be any hyperplane arrangement defined over $\mathbb{Q}$. Then $\mathcal{B}$ is a minor of $\mathcal{A}_n$ for some large enough $n$, that is $\mathcal{B}$ arises from $\mathcal{A}_n$ after a suitable sequence of restriction and contraction steps. Equivalently, any matroid that is r

Figures (2)

  • Figure 1: The resonance arrangement $\mathcal{A}_3$ projected onto the hyperplane $H_{\{1,2,3\}}$. There are $16$ chambers visible and another $16$ antipodal chambers hidden. Thus, $\mathcal{A}_3$ has $32$ chambers in total.
  • Figure 2: Three examples of rectangle circuits together with their side-midpoint tuples.

Theorems & Definitions (39)

  • Definition 1.1
  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Example 1.6
  • Remark 1.7
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • ...and 29 more