A new class of non-identifiable skew-symmetric tensors

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We prove that the generic element of the fifth secant variety $$\sigma _5(Gr(\mathbb {P}^2,\mathbb {P}^9)) \subset \mathbb {P}(\bigwedge ^3 \mathbb {C}^{10})$$ 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 $$(\sigma _3(Gr(\mathbb {P}^2,\mathbb {P}^7)))^{\vee }$$ of the variety of 3-secant planes of the Grassmannian of $$\mathbb {P}^2\subset \mathbb {P}^7$$ 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.