A new class of non-identifiable skew symmetric tensors

Abstract

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.