Tensors with maximal symmetries

Abstract

We classify tensors with maximal and next to maximal dimensional symmetry groups under a natural genericity assumption (1-genericity), in dimensions greater than 7. In other words, we classify minimal dimensional orbits in the space of (m,m,m) tensors assuming 1-genericity. Our study uncovers new tensors with striking geometry. This paper was motivated by Strassen's laser method for bounding the exponent of matrix multiplication. The best known tensor for the laser method is the large Coppersmith-Winograd tensor, and our study began with the observation that it has a large symmetry group, of dimension m^2/2 +m/2. We show that in odd dimensions, this is the largest possible for a 1-generic tensor, but in even dimensions we exhibit a tensor with a larger dimensional symmetry group. In the course of the proof, we classify nondegenerate bilinear forms with large dimensional stabilizers, which may be of interest in its own right.