The Geometry of Polygons in \open R5and Quaternions

Abstract

We consider the moduli space \(\mathcal{M}\)r 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 \(\mathcal{M}\)r and a weighted quotient of n-fold products of the quaternionic projective line \(\mathbb{H}\mathbb{P}\)1 by the diagonal PSL(2, \(\mathbb{H}\))-action. We explore the relation between \(\mathcal{M}\)r 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 \(\mathcal{M}\)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 \(\mathcal{M}\)r.