Planar networks and simple Lie groups beyond type A

TL;DR

This paper extends planar network descriptions to simple Lie groups of types B and C, demonstrating their parametrization, Poisson properties, and connections to cluster coordinates, thus broadening the combinatorial tools for these groups.

Contribution

It introduces network models for types B and C Lie groups, extending known results from type A, and links these models to cluster coordinates and Poisson structures.

Findings

Networks for types B and C are axially symmetric with mutations.

Positive weights parametrize the totally nonnegative parts of these groups.

Constructed network parametrizations of double Bruhat cells and linked face weights to cluster coordinates.

Abstract

The general linear group $GL_{n}$, along with its adjoint simple group $PGL_n$, can be described by means of weighted planar networks. In this paper we give a network description for simple Lie groups of types $B$ and $C$. The corresponding networks are axially symmetric modulo a sequence of cluster mutations along the axis of symmetry. We extend to this setting the result of Gekhtman, Shapiro, and Vainshtein on the Poisson property of Postnikov's boundary measurement map. We also show that $B$ and $C$ type networks with positive weights parametrize the totally nonnegative part of the respective group. Finally, we construct network parametrizations of double Bruhat cells in symplectic and odd-dimensional orthogonal groups, and identify the corresponding face weights with Fock-Goncharov cluster coordinates.