Electrical Networks, Lagrangian Grassmannians, and Symplectic Groups

Moscow Mathematical Journal. 2023. Vol. 23. No. 2. P. 133–167.

B. Bychkov, V. Gorbounov, Talalaev D., Kazakov A.

We refine the result of T. Lam 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 of 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)∩PL, where L⊂⋀n−1R2n−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 Lam's article, define that space L. This connects the combinatorial description of Endiscovered by Lam and representation theory of the symplectic group.

Keywords: Electrical networks, Электрические сети

Publication based on the results of: Representation theory of W-algebras affine and toroidal quantum groups and its applications to geometry, topology and combinatorics (2023)