Circular Planar Electrical Networks II: Positivity Phenomena

TL;DR

This paper explores the algebraic and combinatorial positivity properties of circular planar electrical networks, introducing electrical positroids and analyzing related algebraic structures like Laurent phenomenon algebras.

Contribution

It introduces electrical positroids, providing an axiomatic framework, and investigates the cluster structure of a related Laurent phenomenon algebra.

Findings

Characterization of response matrices via positivity of circular minors.

Introduction of electrical positroids as a new combinatorial object.

Analysis of cluster structures in Laurent phenomenon algebras.

Abstract

Curtis-Ingerman-Morrow characterize response matrices for circular planar electrical networks as symmetric square matrices with row sums zero and non-negative circular minors. In this paper, we study this positivity phenomenon more closely, from both algebraic and combinatorial perspectives. Extending work of Postnikov, we introduce electrical positroids, which are the sets of circular minors which can simultaneously be positive in a response matrix. We give a self-contained axiomatic description of these electrical positroids. In the second part of the paper, we discuss a naturally arising example of a Laurent phenomenon algebra, as studied by Lam-Pylyavskyy. We investigate the clusters in this algebra, building off of initial work by Kenyon-Wilson, using an analogue of weak separation, as was originally introduced by Leclerc-Zelevinsky.