Electrical networks and Lagrangian Grassmannians

Abstract

Cactus networks were introduced by Lam as a generalization of planar electrical networks. He defined a map from these networks to the Grassmannian $\mathrm{Gr}(n+1,2n)$ and showed that the image of this map, ${\mathcal{X}}_n$ lies inside the totally nonnegative part of this Grassmannian. In this paper, we show that ${\mathcal{X}}_n$ is exactly the elements of $\mathrm{Gr}(n+1,2n)$ that are both totally nonnegative and isotropic for a particular skew-symmetric bilinear form. For certain classes of cactus networks, we also explicitly describe how to turn response matrices and effective resistance matrices into points of $\mathrm{Gr}(n+1,2n)$ given by Lam’s map. Finally, we discuss how our work relates to earlier studies of total positivity for Lagrangian Grassmannians.