Electrical networks and hyperplane arrangements

TL;DR

This paper explores Dirichlet arrangements, a generalization of graphic hyperplane arrangements linked to electrical networks and posets, providing new combinatorial descriptions and applications in harmonic functions and order polytopes.

Contribution

It generalizes combinatorial features of graphic arrangements to Dirichlet arrangements, including characteristic polynomials and supersolvability, with applications to electrical networks and order polytopes.

Findings

Generalized characteristic polynomials for Dirichlet arrangements

Established supersolvability of these arrangements

Applied results to harmonic functions and visibility sets

Abstract

This paper studies \emph{Dirichlet arrangements}, a generalization of graphic hyperplane arrangements arising from electrical networks and order polytopes of finite posets. We generalize descriptions of combinatorial features of graphic arrangements to Dirichlet arrangements, including characteristic polynomials and supersolvability. We apply these results to visibility sets of order polytopes and fixed-energy harmonic functions on electrical networks.