Is there a description of the Grassmannian as a homogeneous space where the principal bundle is one associated to the universal vector bundle?

Question

Consider the Grassmannian $\text{Gr}_{n,k}(\mathbb{F})$ for $\mathbb{F}\in \{\mathbb{R},\mathbb{C},\mathbb{H} \}$. Does there exist a presentation of it as a homogeneous space $$ B \hookrightarrow G \to G/B = \text{Gr}_{n,k}(\mathbb{F}) $$ such that the principal $B$-bundle $G \to \text{Gr}_{n,k}(\mathbb{F})$ is the bundle associated to the universal vector bundle on $\text{Gr}_{n,k}(\mathbb{F})$?

Answer 1

Yes there is.

\begin{align*} \operatorname{Gr}_{n,k}(\mathbb{R}) &= O(n)/(O(k)\times O(n-k))\\ \operatorname{Gr}_{n,k}(\mathbb{C}) &= U(n)/(U(k)\times U(n-k))\\ \operatorname{Gr}_{n,k}(\mathbb{H}) &= Sp(n)/(Sp(k)\times Sp(n-k)). \end{align*}

The total space of the tautological bundle is the corresponding Stiefel manifold $V_{n,k}(\mathbb{F})$ which are homogeneous spaces:

\begin{align*} V_{n,k}(\mathbb{R}) &= O(n)/O(n-k)\\ V_{n,k}(\mathbb{C}) &= U(n)/U(n-k)\\ V_{n,k}(\mathbb{H}) &= Sp(n)/Sp(n-k). \end{align*}

Then, for example, we have the bundle $O(k) \to V_{n,k}(\mathbb{R}) \to \operatorname{Gr}_{n,k}(\mathbb{R})$.