The Geometry of Polygons in R^5 and Quaternions

Philip Foth; Guadalupe Lozano

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The Geometry of Polygons in R^5 and Quaternions

Philip Foth·Guadalupe Lozano

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Abstract

We consider the moduli space M_r 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 M_r and a weighted quotient of the n-fold product of the quaternionic projective line HP^1 by the diagonal PSL(2,H)-action. We explore the relation between M_r 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 M_r as a quotient of a subspace in the quaternionic Grassmannian Gr_H(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 M_r.

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Philip Foth, Guadalupe Lozano. 2004-01-23. The Geometry of Polygons in R^5 and Quaternions. https://arxiv.org/abs/math/0202162

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