Electrical Networks, Lagrangian Grassmannians, and Symplectic Groups

We refine the result of T. Lam {L} on embedding the space Eₙ of electrical networks on a planar graph with n boundary points into the totally non-negative Grassmannian Gr≥ 0(n-1,2n) by proving first that the image lands in Gr(n-1,V)⊂ Gr(n-1,2n) where V⊂ R²ⁿ is a certain subspace of dimension $2n-2$. The role of this reduction in the dimension of the ambient space is crucial for us. We show next that the image lands in fact inside the Lagrangian Grassmannian LG(n-1,V)⊂ Gr(n-1,V). As it is well known LG(n-1) can be identified with Gr(n-1,2n-2)∩ P L where L⊂ ⁿ⁻¹ R²ⁿ⁻² is a subspace of dimension equal to the Catalan number Cₙ, moreover it is the space of the fundamental representation of the symplectic group $Sp(2n-2)$ which corresponds to the last vertex of the Dynkin diagram. We show further that the linear relations cutting the image of Eₙ out of Gr(n-1,2n) found in {L} define that space L. This connects the combinatorial description of Eₙ discovered in {L} and representation theory of the symplectic group.