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A new class of non-identifiable skew-symmetric tensors. (English) Zbl 1403.15017

Summary: We prove that the generic element of the fifth secant variety \(\sigma _5(\mathrm{Gr}(\mathbb P^2,\mathbb P^9)) \subset \mathbb P(\bigwedge ^3 \mathbb C^{10})\) 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 \(\mathrm{Gr}(\mathbb P^3, \mathbb P^8)\), is the only non-identifiable case among the non-defective secant varieties \(\sigma _s(\mathrm{Gr}(\mathbb P^k, \mathbb P^n))\) 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 \((\sigma _3(\mathrm{Gr}(\mathbb P^2,\mathbb P^7)))^{\vee }\) of the variety of 3-secant planes of the Grassmannian of \(\mathbb {P}^2\subset \mathbb {P}^7\) is \(\sigma _2(\mathrm{Gr}(\mathbb {P}^2,\mathbb {P}^7))\) 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.

MSC:

15A69 Multilinear algebra, tensor calculus
14M15 Grassmannians, Schubert varieties, flag manifolds
14N05 Projective techniques in algebraic geometry
14L30 Group actions on varieties or schemes (quotients)
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