## 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 ℚ[S_{n} × 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_{n} × S_{n} that arises from the diagonal conjugacy action of S_{n}. More closely, we consider elements of ℚ[S_{n} × 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 ℚ[S_{n} × 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-ε)n^{2}, due to Landsberg, Ottaviani).

Original language | English |
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Pages (from-to) | 457-481 |

Number of pages | 25 |

Journal | Foundations of Computational Mathematics |

Volume | 14 |

Issue number | 3 |

DOIs | |

State | Published - 1 Jan 2014 |

Externally published | Yes |

## Keywords

- Matching polynomial
- Schur ring
- Tensor rank

## ASJC Scopus subject areas

- Analysis
- Computational Mathematics
- Computational Theory and Mathematics
- Applied Mathematics