This is sort of a spinoff of Is there a useful generalization of the Schmidt decomposition to the tensoring together of 3 or more vector spaces? - seems to be almost hopeless, but maybe some partial results are known.
Recall that multivectors are elements of exterior powers: an $n$-dimensional $k$-vector is an element of $\Lambda^k(\mathbb C^n)$. So every multivector is a linear combination of $\it decomposable$ ones - multivectors of the form $v_1\wedge\cdots\wedge v_k$, $v_i\in\mathbb C^n$ and the $\it rank$ of a multivector is the smallest number of summands in such decomposition.
In an answer to the linked question, a decomposition for general tensors is mentioned - some hierarchical higher order singular value decomposition and its generalization called the Tucker decomposition.
From general considerations, such decompositions must be substantially simplifiable if one passes to multivectors. My question is where is it done and what comes out.
Some more specific questions:
1) It is known that $\Lambda^k(V)$ is isomorphic to the space of global sections of a line bundle over the Grassmanian $G_k(V)$ of $k$-dimensional subspaces in $V$ - namely of the highest exterior power of the tautological bundle. Pulling the tautological bundle back to the total space of the projectivization of the bundle splits out of it a canonical 1-dimensional direct summand. Does this correspond to any fact about multivectors? More generally, can the splitting principle be used to produce any decompositions of multivectors?
1') Elements of $\Lambda^k(V)$ are sort of linear combinations of $k$-dimensional subspaces of $V$, so may be viewed as 0-cycles on $G_k(V)$ (with coefficients in the ground field), and the Grassmanian can be located nicely inside of them - see e. g. this answer (essentially this is the formulation of Plücker coordinates). Is this point of view exploited anywhere for the purpose of decomposing multivectors?
2) The case $k=3$ shows clearly why the general question must be quite difficult. Fixing a scalar product on $\mathbb C^n$, 3-vectors may be assigned very special algebra structures on $V$ (in many ways actually), since the scalar product will produce embeddings $\Lambda^3(V)\to\operatorname{Hom}(\Lambda^2(V),V)$. Such algebra structures are special not only because of $x^2=0$ for any $x$ but also the scalar product becomes invariant with respect to them, and $xy$ becomes orthogonal both to $x$ and $y$. So in this case a normal form for trivectors would mean some kind of classification of such algebras - for example, finding bases with the property that products of basis vectors are as small as possible linear combinations of basis vectors. Although very special, such algebras must be very difficult to classify, as they are more general than Lie algebras with an invariant scalar product. Still, classification of simple Lie algebras exists, and they come with their Killing forms. Maybe algebras corresponding to more general trivectors can also be classified somehow? On the other hand, are 3-vectors corresponding to Lie algebra structures in this way characterizable independently regardless of the chosen scalar product? Are there any analogs for higher degree multivectors? (Lie triple systems and alike come to mind...)
