Let $V$ be a real $n$-dimensional vector space, and let $1<k<n,r>1$. I wonder:

Is there a way to characterise which $r$-dimensional subspaces of the exterior power $\bigwedge^k V$ have bases which consist of decomposable elements?

Can we say "how rare" are such subspaces? Do they form a submanifold or an algebraic subvariety? (I think that they do form a closed subset of $Gr_r(\bigwedge^k V)$).

For $r=1$, this amounts to recognizing decomposable elements in $\bigwedge^k V$, which is what the Plucker relations do. But is there any sensible criterion for $r>1$?

Note that a subspace can admit a non-decomposable basis, even when there exist a decomposable basis: Take e.g. $r=2$ and

$\text{span}(e_1 \wedge e_2+e_3 \wedge e_4,e_1 \wedge e_2-e_3 \wedge e_4)=\text{span}(e_1 \wedge e_2,e_3 \wedge e_4)$.

One particular case where a subspace $\tilde W \le \bigwedge^k V$ has a decomposable basis, is when $\tilde W= \bigwedge^k W$ for some subspace $W \le V$. However, I don't know if there is an easy way to recognize subspaces of this form.


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