The geometry of polygons in ℝ5 and quaternions

Philip Foth, Guadalupe Lozano

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

Abstract

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 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(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 languageEnglish (US)
Pages (from-to)209-229
Number of pages21
JournalGeometriae Dedicata
Volume105
Issue number1
DOIs
StatePublished - Apr 2004

Keywords

  • Grassmannians
  • Invariants
  • Lagrangian
  • Polygons
  • Quaternions
  • Reduction

ASJC Scopus subject areas

  • Geometry and Topology

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