Electrical networks and Lagrangian Grassmannians

An electrical network is a graph in a disk with positive
weights on edges. The set of response matrices of a (compactified) set
of electrical networks admits an embedding into the totally
nonnegative Grassmannian $\mathrm{Gr}_{\geq 0}(n-1,2n)$. I will talk
about a new parameterization of the space of electrical networks which
defines an embedding into the Grassmannian $\mathrm{Gr}(n-1,V)$, where
$V$ is a certain subspace of dimension $2n-2$ and moreover an
embedding into the totally nonnegative Lagrangian Grassmannian
$\mathrm{LG}_{\geq 0}(n-1)\subset\mathrm{Gr}(n-1,V).$ The latter
allows us to connect the combinatorics of the space of electrical
networks with the representation theory of the symplectic group. The
talk is based on the joint work with V. Gorbounov, A. Kazakov and D.
Talalaev.