Mathematische Annalen

, Volume 350, Issue 1, pp 79–106

A uniform description of compact symmetric spaces as Grassmannians using the magic square


DOI: 10.1007/s00208-010-0549-8

Cite this article as:
Huang, Y. & Leung, N.C. Math. Ann. (2011) 350: 79. doi:10.1007/s00208-010-0549-8


Suppose \({\mathbb{A}}\) and \({\mathbb{B}}\) are normed division algebras, i.e. \({\mathbb{R}, \mathbb{C}, \mathbb{H}}\) or \({\mathbb{O}}\), we introduce and study Grassmannians of linear subspaces in \({(\mathbb{A}\otimes\mathbb{B})^{n}}\) which are complex/Lagrangian/maximal isotropic with respect to natural two tensors on \({(\mathbb{A}\otimes\mathbb{B})^{n}}\). We show that every irreducible compact symmetric space must be one of these Grassmannian spaces, possibly up to a finite cover. This gives a simple and uniform description of all compact symmetric spaces. This generalizes the Tits magic square description for simple Lie algebras to compact symmetric spaces.

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.The Department of MathematicsJinan UniversityGuangzhouChina
  2. 2.The Institute of Mathematical SciencesThe Chinese University of Hong KongShatinHong Kong