On cap sets and the group-theoretic 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 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.
- Publication:
-
arXiv e-prints
- 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
- E-Print:
- 27 pages