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Polygon spaces and Grassmannians. (English) Zbl 0888.58007
Summary: We study the moduli spaces of polygons in \(\mathbb{R}^2\) and \(\mathbb{R}^3\), 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.

58D29 Moduli problems for topological structures
14M15 Grassmannians, Schubert varieties, flag manifolds
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems