The Bruhat decomposition of real Grassmann manifolds

TL;DR

This paper investigates the Bruhat decomposition of real Grassmann manifolds, establishing a CW complex structure and calculating incidence numbers, thereby enhancing understanding of their geometric and topological properties.

Contribution

It introduces a CW structure on Grassmann manifolds derived from Bruhat decomposition and computes the incidence numbers between cells.

Findings

Established a CW complex structure on Grassmann manifolds

Determined incidence numbers between cells

Connected Bruhat decomposition to topological cell structures

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 $Sl_n({\mathbb R})$ 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.