We consider the moduli space Mr of polygons with fixed side lengths in five-dimensional Euclidean space. We analyze the local structure of its singularities and exhibit a real-analytic equivalence between Mr and a weighted quotient of n-fold products 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 Grℂ(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 M r 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 marices and briefly describe generalized action-angle coordinates on Mr.
|Original language||English (US)|
|Number of pages||21|
|State||Published - Apr 2004|
ASJC Scopus subject areas
- Geometry and Topology