The Bruhat decomposition of real Grassmann manifolds

Abstract

We study the Grassmann manifold $G_k$ of all $k$-dimensional subspaces of ${\mathbb{R}}^n$. The Cartan embedding $G_k\subset O(n)$ realizes $G_k$ as a subspace of $S{l}_n(\mathbb{R})$ and we study the decomposition $G_k={\coprod }_w(BwB\cap G_k)$ inherited from the classical Bruhat decomposition. We prove that this defines a CW structure on $G_k$ and determine the incidence numbers between cells.

References in corpus (1)

• Governing Singularities of Schubert Varieties