My question may be absolutely elementary and is probably answered in 19th century. A reference or a short clear argument would be highly appreciated.


Let $V\_1, \ldots V\_n$ be finite dimensional vector spaces over the same field (may assume complex numbers). What are $GL(V\_1)\times \ldots \times GL(V\_n)$-orbits on $V\_1 \otimes \ldots \otimes V\_n$?


The only invariant of an orbit I can see is "a multirank" $(k\_1, \ldots k\_n)$ where $k\_i$ is the dimension of support of an element in $V\_i$. The multirank satisfies inequalities $k\_i \leq \prod\_{j\neq i} k\_j$. Would it be too naive to suggest that orbits are in 1-1 correspondence with legal multiranks?



Invariants and orbits of $n$-tensors
Here is a start, suppose that $V\_i$ is $\mathbb C^{k\_i}$ (and restricting to $k\_1,k\_2,\dots,k\_n, n\geq 2$). The tuples $(k\_1,k\_2,\dots,k\_n)$ for which the action of $GL\_{k\_1}\times\cdots\times GL\_{k\_n}$ on $\mathbb{C}^{k\_1}\otimes \cdots\otimes \mathbb{C}^{k\_n}$ has only finitely many orbits are $(k,l),(2,2,k),(2,3,k)$, for positive integers $k,l$. This was proven in 



> 
> V. G. Kac, "Some remarks on nilpotent orbits", J. Algebra 64 (1980), 190–213.
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These orbits are classified in "Orbits and their closures in the spaces $\mathbb{C}^{k\_1}\otimes \cdots\otimes \mathbb{C}^{k\_r}$" by P.G. Parfenov ([MR](http://www.ams.org/mathscinet-getitem?mr=1830474)). This paper doesn't seem to be freely online in English, but Russian version is [here](http://m.mathnet.ru/links/aef8cfc03fa5000520d5774ed70c45cf/sm537.pdf), and I believe you can find a summary in section 5 [here](http://arxiv.org/abs/0909.4262).


