Polygon spaces and Grassmannians

1996/02/29 by Jean-Claude Hausmann, Allen Knutson, Hausmann, Jean-Claude +1 · 5 citations
Computer Science · Mathematics · #Computational Geometry and Mesh Generation #Differential Geometry (math.DG) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.dg-ga/9602012

openalex publication_date 1996/02/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the moduli spaces of polygons in R² and R³, identifying them with subquotients of 2-Grassmannians using a symplectic version of the Gel'fand-MacPherson correspondence. We show that the bending flows defined by Kapovich-Millson arise as a reduction of the Gel'fand-Cetlin system on the Grassmannian, and with these determine the pentagon and hexagon spaces up to equivariant symplectomorphism. Other than invocation of Delzant's theorem, our proofs are purely polygon-theoretic in nature.

Cited by

• Spin actions and Polygon spaces
• On the GIT quotient of Grassmannians by one dimensional torus
• Semitoric Families on Pentagon Spaces
• The tetrahedral Horn problem and asymptotics of U(n) 6j symbols
• Smooth manifolds in G_n,2 and ℂ P^N defined by symplectic reductions of Tⁿ-action

Related

• Singular fibers of the bending flows on the moduli space of 3D polygons
• The Geometry of Polygons in R⁵ and Quaternions
• The Symplectic Geometry of Polygons in the 3-sphere
• The symplectic geometry of polygons in hyperbolic 3-space
• Moduli Spaces of Stable Polygons and Symplectic Structures on M_0,n