Abelian Tensors

TL;DR

This paper investigates tensors satisfying Strassen's equations, providing geometric characterizations of key tensors, reducing the problem to symmetric tensors, and exploring related algebraic varieties.

Contribution

It offers new geometric characterizations of the Coppersmith-Winograd tensor and links tensor properties to abelian subspace varieties and reductions.

Findings

Characterization of Coppersmith-Winograd tensor

Reduction to symmetric tensor study under genericity

Connections to abelian subspace varieties

Abstract

We analyze tensors in the tensor product of three m-dimensional vector spaces satisfying Strassen's equations for border rank m. Results include: two purely geometric characterizations of the Coppersmith-Winograd tensor, a reduction to the study of symmetric tensors under a mild genericity hypothesis, and numerous additional equations and examples. This study is closely connected to the study of the variety of m-dimensional abelian subspaces of the space of endomorphisms of an m-dimensional vector space, and the subvariety consisting of the Zariski closure of the variety of maximal tori, called the variety of reductions.