Construction of Grassmann manifolds

Question

Is there a way to construct the Grassmann manifold via block matrices? For example the upper triangular matrices stabilize the (coordinate) basis of $\mathbb R^n$.

Answer 1

You can read a construction of the Grassmanianns $G(k,n)$, over an algebraically closed field $\mathbb{K}$, as you required in http://www.math.uchicago.edu/~may/VIGRE/VIGRE2007/REUPapers/FINALFULL/Hudec.pdf; for exact, the proposition 2.4 at page 4.

UPDATE!

Let $\mathbb{V}$ be a real vector space of dimension $n$, let $k\in\{1,\dots,n-1\}$, let $G(k,n)$ be the set of all $k$-planes of $\mathbb{V}$ and let $\{e_1,\dots,e_n\}$ be a basis of $\mathbb{V}$; defined \begin{equation} \forall M\in\mathrm{GL}(n,\mathbb{R}),\,M\cdot\langle e_1,\dots,e_k\rangle=\langle Me_1,\dots,Me_k\rangle\in G(k,n), \end{equation} one can consider the action: \begin{equation} \alpha:(M,W)\in\mathrm{GL}(n,\mathbb{R})\times G(k,n)\to M\cdot W\in G(k,n). \end{equation} One can prove that $\alpha$ is a transitive action, and it turns out that $G(k,n)$ is in bijection with \begin{equation} \mathrm{GL}(n,\mathbb{R})_{\displaystyle/P(k)}, \end{equation} where $P(k)$ is the (closed) subgroup of $\mathrm{GL}(n,\mathbb{R})$ generated by the matrices of the following type: \begin{equation} A\in\mathbb{R}_k^k,B\in\mathbb{R}_k^{n-k},C\in\mathbb{R}_{n-k}^{n-k},\begin{pmatrix} A & B\\ \underline{0}_{n-k}^k & C \end{pmatrix}. \end{equation} For other details, you can consult Warner - Foundations of Differentiable Manifolds and Lie Groups, chapter 3.