On cap sets and the group-theoretic approach to matrix multiplication

TL;DR

This paper investigates the group-theoretic approach to matrix multiplication, demonstrating limitations when using abelian groups of bounded exponent, and connecting tensor rank concepts to geometric invariant theory.

Contribution

It rules out the possibility of achieving matrix multiplication exponent two via abelian groups of bounded exponent within the existing framework.

Findings

Bound the size of tricolored sum-free sets in abelian groups of bounded exponent.

Extend results on cap sets to this new setting.

Link tensor rank variants to unstable tensors in geometric invariant theory.

Abstract

In 2003, Cohn and Umans described a framework for proving upper bounds on the exponent $\omega$ of matrix multiplication by reducing matrix multiplication to group algebra multiplication, and in 2005 Cohn, Kleinberg, Szegedy, and Umans proposed specific conjectures for how to obtain $\omega=2$. In this paper we rule out obtaining $\omega=2$ in this framework from abelian groups of bounded exponent. To do this we bound the size of tricolored sum-free sets in such groups, extending the breakthrough results of Croot, Lev, Pach, Ellenberg, and Gijswijt on cap sets. As a byproduct of our proof, we show that a variant of tensor rank due to Tao gives a quantitative understanding of the notion of unstable tensor from geometric invariant theory.