Orbits of exterior products

Question

In linear algebra one learns a lot of normal forms (Which I want to think of as a classification of the orbits of a group action on some set). For example if $V$ is a $k$-vectorspace $GL(V)$ acts on $V$ with exactly two orbits - the orbit of $0$ and the other orbit.

Now what happens if we let $GL(V)$ act diagonally on $V^{\wedge n}$ or $V^{\otimes n}$? Can one give a normal form for this group action or at least find the number of orbits ?

The motivation for this question comes from surgery theory. The number of isomorphism types of fake $n$-tori (for $n\ge 5$) is given by the number of orbits of $GL_n(\mathbb{F}_2)$ acting on $(\mathbb{F}_2^n)^{\wedge n-3}$.

Answer 1

To the best of my knowledge, already for $n=3$ case (which probably is close, if not identical, to $n=\dim(V)-3$ because of the duality argument) the situation is far from clear for $\dim(V)$ starting from $10$ or so (the complexity somewhat depends on the ground field). Some hints on that are in papers like [1], [2], [3], and many others which you find in Mathscinet or such asking for "classification of trivectors".

Of course, for $n=2$ the classification is straightforward.