Tensor Rank: Matching Polynomials and Schur Rings

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 5n²/4 on the border rank of the matrix multiplication tensor (being, of course, weaker than the best known one (2-ε)n², due to Landsberg, Ottaviani).