Planar networks and simple Lie groups beyond type A
Year: 2025
Type: article
Abstract: The general linear group G L n , along with its adjoint simple group P G L 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.
Source: Advances in Mathematics
Author: Anton Izosimov
Institutions: University of Arizona, University of Glasgow
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,790
OpenAlex ID: W4413318485