Electrical networks, Lagrangian Grassmannians, and symplectic groups

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.