Moscow Mathematical Journal

Volume 23, Issue 2, April–June 2023 pp. 133–167.

Electrical Networks, Lagrangian Grassmannians, and Symplectic Groups

Authors: B. Bychkov (1), V. Gorbounov (2), A. Kazakov (3), and D. Talalaev (4)
Author institution:(1) Department of Mathematics, University of Haifa, Mount Carmel, 3488838, Haifa, Israel;
Faculty of Mathematics, National Research University Higher School of Economics, Usacheva 6, 119048 Moscow, Russia;
Centre of Integrable Systems, P. G. Demidov Yaroslavl State University, Sovetskaya 14, 150003, Yaroslavl, Russia
(2) Faculty of Mathematics, National Research University Higher School of Economics, Usacheva 6, 119048 Moscow, Russia
(3) Lomonosov Moscow State University, Moscow, Russia;
Center of Fundamental Mathematics, Moscow Institute of Physics and Technology (National Research University), Russia;
Centre of Integrable Systems, P. G. Demidov Yaroslavl State University, Sovetskaya 14, 150003, Yaroslavl, Russia
(4) Lomonosov Moscow State University, Moscow, Russia;
Centre of Integrable Systems, P. G. Demidov Yaroslavl State University, Sovetskaya 14, 150003, Yaroslavl, Russia

Summary:

We refine the result of T. Lam on embedding the space $E_n$ of electrical networks on a planar graph with $n$ boundary points into the totally non-negative Grassmannian $\mathrm{Gr}_{\geq 0}(n-1,2n)$ by proving first that the image lands in $\mathrm{Gr}(n-1,V)\subset \mathrm{Gr}(n-1,2n)$, where $V\subset \mathbb{R}^{2n}$ is a certain subspace of dimension $2n-2$. The role of this reduction of the dimension of the ambient space is crucial for us. We show next that the image lands in fact inside the Lagrangian Grassmannian $\mathrm{LG}(n- 1,V)\subset \mathrm{Gr}(n-1,V)$. As it is well known $\mathrm{LG}(n-1)$ can be identified with $\mathrm{Gr}(n- 1,2n-2)\cap \mathbb{P} L$, where $L\subset \bigwedge^{n-1}\mathbb R^{2n-2}$ is a subspace of dimension equal to the Catalan number $C_n$, moreover it is the space of the fundamental representation of the symplectic group $\mathrm{Sp}(2n-2)$ which corresponds to the last vertex of the Dynkin diagram. We show further that the linear relations cutting the image of $E_n$ out of $\mathrm{Gr}(n-1,2n)$, found in Lam's article, define that space $L$. This connects the combinatorial description of $E_n$ discovered by Lam and representation theory of the symplectic group.

2020 Math. Subj. Class. 14M15, 82B20, 05E10.

Keywords: Electrical networks, Electrical Algebra, Lagrangian Grassmanian.

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