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Article Dans Une Revue Foundations of Computational Mathematics Année : 2014

TENSOR RANK: MATCHING POLYNOMIALS AND SCHUR RINGS

Dima Grigoriev
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Mikhail Muzychuk
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Ilya Ponomarenko
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Résumé

We study the polynomial equations vanishing on tensors of a given rank. By means of polarization we reduce them to elements A of the group algebra Q[Sn × Sn] and describe explicit linear equations on the coefficients of A to vanish on tensors of a given rank. Further, we reduce the study to the Schur ring over the group Sn × Sn that arises from the diagonal conjugacy action of Sn. More closely, we consider elements of Q[Sn × Sn] vanishing on tensor of rank n − 1 and describe them in terms of triples of Young diagrams, their irreducible characters and nonvanishing of their Kronecker coefficients. Also, we construct a family of elements in Q[Sn × Sn] vanishing on tensors of rank n − 1 and illustrate our approach by a sharp lower bound on the border rank of an explicitly produced tensor. Finally, we apply this construction to prove a lower bound 5n 2 /4 on the border rank of the matrix multiplication tensor (being, of course, weaker than the best known one (2 −) • n 2 , due to Landsberg, Ottaviani).
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hal-03044846 , version 1 (07-12-2020)

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  • HAL Id : hal-03044846 , version 1

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Dima Grigoriev, Mikhail Muzychuk, Ilya Ponomarenko. TENSOR RANK: MATCHING POLYNOMIALS AND SCHUR RINGS. Foundations of Computational Mathematics, 2014. ⟨hal-03044846⟩

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