On the structure tensor of sln

Abstract

The structure tensor of sln is the tensor arising from the Lie bracket bilinear operation on the set of traceless n by n complex matrices. This tensor is intimately related to the well studied matrix multiplication tensor. Studying the structure tensor of sln may provide further insight into the complexity of matrix multiplication and the ``hay in a haystack'' problem of finding explicit sequences tensors with high rank or border rank. We aim to find new bounds on the rank and border rank of this structure tensor when n is 3 and 4. The lower bounds on the border rank in the case of sl4 were obtained via koszul flattenings and border substitution. The best lower bound on the border rank in the case of sl3 were obtained via a new technique called border apolarity, developed by Conner, Harper, and Landsberg. Upper bounds on the rank of the structure tensor of sl3 are obtained via numerical methods that allowed us to find an explicit rank decomposition.