# Non-(stable)-triviality of the tautological bundles

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 $\gamma_k(\mathbb{K}^N)$ is never stably trivial unless $k = 0, N$. But I don't know how to prove this. Maybe by a more elaborate argument using the Stiefel-Whitney or Chern classes? But then, what about $\gamma_k(\mathbb{H}^N)$, which do not have associated characteristic classes?

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Posted this question on MO: mathoverflow.net/questions/131340 –  AlexE May 21 '13 at 13:37