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.

The article has been accepted by Annali di Matematica Pura ed Applicata

References in corpus (1)

• Exact and Asymptotic Measures of Multipartite Pure State Entanglement

Cited by in corpus (1)

• On complex and real identifiability of tensors