Electrical networks and hyperplane arrangements (Q2326914)

In this nice paper, the author introduces and studies a new class of hyperplane arrangements, the so-called Dirichlet arrangements, arising from electrical networks and order polytopes of finite posets. Let \(G=(V,E)\) be a graph, then the graphic arrangement \(\mathcal{A}_{G}\) is the set of hyperplanes in \(\mathbb{R}^{V}\) given by \(x_{i}=x_{j}\) for all edges \(ij \in E\). Let \(\Gamma\) be a finite connected undirected graph with no loops or multiple edges. Let \(\partial V \subset V\) be a set of at least \(2\) vertices called boundary nodes, no two of which are adjacent. Let \(u : \partial V \rightarrow \mathbb{R}\) be injective. The Dirichlet arrangement \(\mathcal{A}_{\Gamma, u}\) is the set of intersections of hyperplanes in the graphic arrangement \(\mathcal{A}_{\Gamma}\) with the affine subspace \[ \{x \in \mathbb{R}^{V} : x_{j} = u(j) \text{ for all } j \in \partial V\}. \] It is worth emphasizing that the Dirichlet arrangement \(\mathcal{A}_{\Gamma, u}\) is not a restriction arrangement in the traditional sense. The first main result of the paper generalizes a key theorem on graphic arrangement. Theorem 1. Let \(\widehat{\Gamma}\) be the graph obtained from \(\Gamma\) by adding an edge between each pair of boundary nodes. The following hold: i) The intersection poest \(L(\mathcal{A}_{\Gamma, u})\) is the order ideal of \(L(\mathcal{A}_{\Gamma})\) consisiting of all boundary-separating connected partitions of \(\Gamma\). ii) The characteristic polynomial of \(\mathcal{A}_{\Gamma, u}\) is the quotient of the chromatic polynomial of \(\widehat{\Gamma}\) by a falling factor. iii) If \(\widehat{\Gamma}\) is \(2\)-connected, then the boundary chambers of \(\mathcal{A}_{\Gamma, u}\) correspond to the possible orientations of current flow through \(\Gamma\) respecting the voltages \(u\) and in which the current flowing through each edge is non-zero. Let us recall a theorem by Stanley which tells us that a graphic arrangement \(\mathcal{A}_{\Gamma}\) is supersolvable iff \(\Gamma\) is chordal. A very similar result holds for Dirichlet arrangements. Theorem 2. The Dirichlet arrangement \(\mathcal{A}_{\Gamma, u}\) is supersolvable iff the graph \(\widehat{\Gamma}\) as in Theorem 1 is chordal. Now we focus on some applications. Let \(\mathcal{P}\) be a finite poset and \(\mathcal{O}(\mathcal{P})\) be the convex polytope in \(\mathbb{R}^{\mathcal{P}}\) of all order-preserving functions \(\mathcal{P} \rightarrow [0,1]\). The polytope \(\mathcal{O}(\mathcal{P})\) is called the order polytope of \(\mathcal{P}\). We consider the set of facets of \(\mathcal{O}(\mathcal{P})\) visible from different points in \(\mathbb{R}^{\mathcal{P}}\), the visibility set. Finally, for a graph \(\Gamma\) we denote by \(\alpha(\Gamma)\) the number of acyclic orientations and by \(\beta(\Gamma)\) the beta invariant. Proposition 1. Let \(\mathcal{P}\) be a finite poset. There is a graph \(G\) such that \(\mathcal{O}(\mathcal{P})\) has exactly \(\frac{1}{2}\alpha(G)\) visibility sets of which exactly \(\frac{1}{2} \alpha(G) - \beta(G)\) are visible from far away. The last part of the paper is devoted to applications involving electrical networks with fixed boundary voltages (for instance \(\varepsilon\)-harmonic functions).