Spin actions and Polygon spaces

Abstract

In this article, we construct correspondences between polygon spaces in Euclidean spaces of dimension $2,3,5,9$and the quotient spaces of $2$-Steifel manifolds along the normed division algebra$\mathbb{F}$ real $\mathbb{R}$, complex $\mathbb{C}$, quaternions $\mathbb{H}$, octonions $\mathbb{O}$. For the purpose, we introduce Hopf map on ${\mathbb{F}}^2$and consider the spin action of $SU\left(2,\mathbb{F}\right)$ to spinor ${\mathbb{F}}^2$and the induced $SO$action to the Euclidean space $\mathbb{R}\oplus \mathbb{F}$. The correspondences are extension of the work of Hausmann and Knutson for polygon spaces of dimension $2,3$and $2$-Grassmannians over real and complex.

32 pages