Take the 2-minute tour ×  This is a question I asked at Math.SE
but got no answers: http://math.stackexchange.com/q/396217/7110/
The tautological vector bundle $\gamma_k(\mathbb{K}^N)$ over the
Grassmann manifold $G_k(\mathbb{K}^N)$ of all $k$-planes in
$\mathbb{K}^N$ (for $\mathbb{K} = \mathbb{R}$, $\mathbb{C}$ or
$\mathbb{H}$) is defined as $\gamma_k(\mathbb{K}^N) := \{(x, v)
\in G_k(\mathbb{K}^N) \times \mathbb{K}^N: v \in x\}$. I
wondered whether any of those bundles is (stably) trivial:  For
which $k$, $N$ is $\gamma_k(\mathbb{K}^N)$ (stably) trivial?
Sure it is the case for $k = 0$ or $k = N$ since then
$G_k(\mathbb{K}^N)$ is just a point. Furthermore, I know that
$\gamma_1(\mathbb{R}^2)$ is not stably trivial since it is the
Möbius bundle which has non-trivial first Stiefel-Whitney class
$\omega_1(\gamma_1(\mathbb{R}^2)) \not= 0$. Similar, since
$c_1(\gamma_1(\mathbb{C}^N)) \not= 0$ (more or less per
definition), we know that $\gamma_1(\mathbb{C}^N)$ is not stably
trivial.  I think that $\gamma_k(\ma