Jonah Blasiak, Thomas Church, Henry Cohn, Joshua A. Grochow, Chris Umans

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

http://arxiv.org/abs/1605.06702

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. In 2005 Cohn, Kleinberg, Szegedy, and Umans proposed specific conjectures for how to obtain ω=2 in this framework. In this note we rule out obtaining ω=2 in this framework from the groups (𝔽_p)^n, using the breakthrough results of Croot, Lev, Pach, Ellenberg, and Gijswijt on cap sets. These restrictions do not however rule out abelian groups in general, let alone nonabelian groups.

]]>Let .

Define to be the subspace of functions in , that satisfy for all .

Then .

Define a symmetric bilinear form on elements of by .

Since , for all , we have ,

which gives us sine for all .

Thus, is contained in its orthogonal complement, which implies that .

This gives , implying . ]]>

I tried to type this as a comment in Terence Tao’s blog, but it looks very ugly after formatting. Thus pdf.

Fedya

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