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[Sn×Sn] and describe explicitely 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 5n2/4 on the border rank of the matrix multiplication tensor (being, of course, weaker than the best known one 3n2/2+n/2−1, due to T.Lickteig).