, Volume 350, Issue 1, pp 79-106
Date: 30 Jul 2010

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

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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.