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arXiv · 2106.15418

Electrical networks and Lagrangian Grassmannians

Sunita Chepuri·Terrence George·David E Speyer
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Abstract

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, $\mathcal X_n$ lies inside the totally nonnegative part of this Grassmannian. In this paper, we show that $\mathcal X_n$ is exactly the elements of 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.

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Sunita Chepuri, Terrence George, David E Speyer. 2021-06-29. Electrical networks and Lagrangian Grassmannians. https://arxiv.org/abs/2106.15418

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