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arXiv · 1605.06702

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

Jonah Blasiak·Thomas Church·Henry Cohn·Joshua A. Grochow·Eric Naslund·William F. Sawin·Chris Umans
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Abstract

In 2003, Cohn and Umans described a framework for proving upper bounds on the exponent $ω$ 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 $ω=2$. In this paper we rule out obtaining $ω=2$ 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.

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Jonah Blasiak, Thomas Church, Henry Cohn, Joshua A. Grochow, Eric Naslund, William F. Sawin, Chris Umans. 2017-01-14. On cap sets and the group-theoretic approach to matrix multiplication. https://doi.org/10.19086/da.1245

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