2002/02/17 by Philip Foth, Foth, Philip, Guadalupe Lozano +1
Mathematics · #Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics #Symplectic Geometry (math.SG) #math.AG #math.DG #math.SG
paper · pdf · doi:10.48550/arxiv.math/0202162
22 pages
arxiv created 2004/01/23 · arxiv updated 2009/11/30
We consider the moduli space Mr of polygons with fixed side lengths in five-dimensional eucledian space. We analyze the local structure of its singularities and exhibit a real-analytic equivalence between Mr and a weighted quotient of the n-fold product of the quaternionic projective line HP1 by the diagonal PSL(2,H)-action. We explore the relation between Mr and the fixed point set of an anti-symplectic involution on a GIT quotient Gr(2,4)n/SL(4,C). We generalize the Gel'fand-MacPherson correspondence to more general complex Grassmannians and to the quaternionic context, and realize our space Mr as a quotient of a subspace in the quaternionic Grassmannian GrH(2,n) by the action of the group Sp(1)n. We also give analogues of the Gel'fand-Tsetlin coordinates on the space of quaternionic Hermitean matrices and briefly describe generalized action-angle coordinates on Mr.