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Electrical Networks, Lagrangian Grassmannians and Symplectic Groups

Boris Bychkov, Vassily Gorbounov, Anton Kazakov, Dmitry Talalaev

Abstract

We refine the result of T. Lam \cite{L} 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 in 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 $Sp(2n-2)$ 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 \cite{L} define that space $L$. This connects the combinatorial description of $E_n$ discovered in \cite{L} and representation theory of the symplectic group.

Electrical Networks, Lagrangian Grassmannians and Symplectic Groups

Abstract

We refine the result of T. Lam \cite{L} on embedding the space of electrical networks on a planar graph with boundary points into the totally non-negative Grassmannian by proving first that the image lands in where is a certain subspace of dimension . 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 . As it is well known can be identified with where is a subspace of dimension equal to the Catalan number , moreover it is the space of the fundamental representation of the symplectic group which corresponds to the last vertex of the Dynkin diagram. We show further that the linear relations cutting the image of out of found in \cite{L} define that space . This connects the combinatorial description of discovered in \cite{L} and representation theory of the symplectic group.

Paper Structure

This paper contains 19 sections, 38 theorems, 120 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Organisation of the paper
  3. Electircal networks
  4. Postnikov's approach and Lam's embedding
  5. Electrical networks
  6. Electrical networks and Grassmannian
  7. New parametrisation
  8. Compactification
  9. Lagrangian Grassmannian
  10. Fundamental representation of the symplectic group
  11. The Lam group
  12. The symplectic group and the Lagrangian Grassmannian
  13. Non-negative Lagrangian Grassmannian
  14. Applications of the new parametrisation
  15. Adjoining edges and the change of the response matrix
  16. ...and 4 more sections

Key Result

Theorem 2.5

LT For a bipartite network $N(\Gamma, \omega)$ the collection of boundary measurements $\Delta_{I}^M$ considered as a set of Plucker coordinates defines a point in the non-negative Grassmannian $\mathrm{Gr}(k(\Gamma), n)$. In particular the boundary measurements $\Delta_{I}^M$ satisfy the Plucker re

Figures (12)

  • Figure 1: Grove measurements and Plucker coordinates
  • Figure 2: Example \ref{['ex']}
  • Figure 3: Adding a boundary bridge and a boundary spike
  • Figure 4: Adding a bridge
  • Figure 5: Electrical and its bipartite networks
  • ...and 7 more figures

Theorems & Definitions (87)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • Definition 2.6
  • Remark 2.7
  • Definition 2.8
  • Proposition 2.9
  • Definition 2.10
  • ...and 77 more