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- W2052520425 abstract "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 $${mathbb {Q}}[S_ntimes S_n]$$ 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 $$S_ntimes S_n$$ that arises from the diagonal conjugacy action of $$S_n$$ . More closely, we consider elements of $${mathbb {Q}}[S_ntimes S_n]$$ vanishing on tensors 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 $${mathbb {Q}}[S_ntimes S_n]$$ 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 $$(2-epsilon )cdot n^2$$ , due to Landsberg, Ottaviani)." @default.
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- W2052520425 date "2014-03-25" @default.
- W2052520425 modified "2023-09-25" @default.
- W2052520425 title "Tensor Rank: Matching Polynomials and Schur Rings" @default.
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- W2052520425 doi "https://doi.org/10.1007/s10208-014-9190-3" @default.
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