  We establish basic information about border rank algorithms for the matrix
multiplication tensor and other tensors with symmetry. We prove that border
rank algorithms for tensors with symmetry (such as matrix multiplication and
the determinant polynomial) come in families that include representatives with
normal forms. These normal forms will be useful both to develop new efficient
algorithms and to prove lower complexity bounds. We derive a border rank
version of the substitution method used in proving lower bounds for tensor
rank. We use this border-substitution method and a normal form to improve the
lower bound on the border rank of matrix multiplication by one, to 2n^2- n+1.
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.

Algebraic Geometry;Computational Complexity