A new class of non-identifiable skew symmetric tensors

We prove that the generic element of the fifth secant variety $sigma_5(Gr(mathbb{P}^2,mathbb{P}^9)) subset mathbb{P}(igwedge^3 mathbb{C}^{10})$ of the Grassmannian of planes of $mathbb{P}^9$ has exactly two decompositions as a sum of five projective classes of decomposable skew-symmetric tensors. {We show that this, {together with $Gr(mathbb{P}^3, mathbb{P}^8)$, is the only non-identifiable case} among the non-defective secant varieties $sigma_s(Gr(mathbb{P}^k, mathbb{P}^n))$ for any $n<14$. In the same range for $n$, we classify all the weakly defective and all tangentially weakly defective secant varieties of any Grassmannians.} We also show that the dual variety $(sigma_3(Gr(mathbb{P}^2,mathbb{P}^7)))^{ee}$ of the variety of 3-secant planes of the Grassmannian of $mathbb{P}^2subset mathbb{P}^7$ is $sigma_2(Gr(mathbb{P}^2,mathbb{P}^7))$ the variety of bi-secant lines of the same Grassmannian. The proof of this last fact has a very interesting physical interpretation in terms of measurement of the entanglement of a system of 3 identical fermions, the state of each of them belonging to a 8-th dimensional ``Hilbert'' space.