# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1703.09276