On the Geometry of Border Rank Decompositions for Matrix Multiplication and Other Tensors with Symmetry

We present a new approach to study tensors with symmetry, via local algebraic geometry. Border rank decompositions for such tensors---in particular, matrix multiplication and the determinant polynomial---come in families. We prove that these families include representatives with normal forms. These normal forms will be useful to prove lower complexity bounds and possibly even to determine new decompositions. We derive a border rank version of the substitution method used in proving lower bounds for tensor rank. Applying these methods, we improve the lower bound on the border rank of matrix multiplication. We also point out difficulties that will be formidable obstacles to future progress on lower complexity bounds for tensors because of the “wild” structure of the Hilbert scheme of points.