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JOURNALS // Moscow Mathematical Journal // Archive

Mosc. Math. J., 2023 Volume 23, Number 2, Pages 133–167 (Mi mmj850)

Electrical networks, Lagrangian Grassmannians, and symplectic groups

B. Bychkovabc, V. Gorbounovb, A. Kazakovdec, D. Talalaevdc

a Department of Mathematics, University of Haifa, Mount Carmel, 3488838, Haifa, Israel
b Faculty of Mathematics, National Research University Higher School of Economics, Usacheva 6, 119048 Moscow, Russia
c Centre of Integrable Systems, P. G. Demidov Yaroslavl State University, Sovetskaya 14, 150003, Yaroslavl, Russia
d Lomonosov Moscow State University, Moscow, Russia
e Center of Fundamental Mathematics, Moscow Institute of Physics and Technology (National Research University), Russia

Abstract: 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.

Key words and phrases: electrical networks, electrical algebra, Lagrangian Grassmanian.

MSC: 14M15, 82B20, 05E10

Language: English



© Steklov Math. Inst. of RAS, 2024