Electrical networks and Lagrangian Grassmannians

Cactus networks were introduced by Lam as a generalization of planar electrical networks. He defined a map from these networks to the Grassmannian Gr(n+1,2n) and showed that the image of this map, Xₙ lies inside the totally nonnegative part of this Grassmannian. In this paper, we show that Xₙ is exactly the elements of 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 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.