Polygon spaces and Grassmannians

TL;DR

This paper explores the geometric structure of polygon moduli spaces in two and three dimensions, linking them to Grassmannians and symplectic geometry, and clarifies their integrable systems and symplectic reductions.

Contribution

It establishes a novel connection between polygon spaces and Grassmannians via symplectic reduction, providing polygon-theoretic proofs for their symplectic structures.

Findings

Polygon spaces are identified with subquotients of Grassmannians.

Bending flows correspond to reductions of the Gel'fand-Cetlin system.

Polygon spaces like pentagon and hexagon are characterized up to symplectomorphism.

Abstract

We study the moduli spaces of polygons in R^2 and R^3, 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.