Planar networks and simple Lie groups beyond type A

2023/08/27 by Anton Izosimov, Izosimov, Anton
Mathematics · #Advanced Algebra and Geometry #Combinatorics (math.CO) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT) #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.2308.13975

openalex publication_date 2023/08/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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