On cap sets and the grouptheoretic approach to matrix multiplication
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 sumfree 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.
 Publication:

arXiv eprints
 Pub Date:
 May 2016
 DOI:
 10.48550/arXiv.1605.06702
 arXiv:
 arXiv:1605.06702
 Bibcode:
 2016arXiv160506702B
 Keywords:

 Mathematics  Combinatorics;
 Computer Science  Data Structures and Algorithms;
 Mathematics  Group Theory
 EPrint:
 27 pages