A new class of non-identifiable skew-symmetric tensors

We prove that the generic element of the fifth secant variety sigma(5)(Gr (P-2 , P-9 )) subset of P(Lambda(3) C-10) of the Grassmannian of planes of 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 (P-3 , P-8 ), is the only non-identifiable case among the non-defective secant varieties sigma(s) (Gr (P-k , 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 (P-2 , P-7 )))(v) of the variety of 3-secant planes of the Grassmannian of P-2 subset of P-7 is sigma(2)(Gr (P-2 , 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.