A Note on the Group-theoretic Approach to Fast Matrix Multiplication

TL;DR

This paper refines the group-theoretic approach to fast matrix multiplication by simplifying assumptions on subsets and providing a shorter proof for an important upper bound related to the Triple Product Property.

Contribution

It shows that subsets S, T, U can include the identity and be disjoint, and offers a more concise proof of the key upper bound involving their sizes.

Findings

Subsets S, T, U can contain the identity element.

S, T, U can be assumed disjoint.

Established a shorter proof of the bound |S|+|T|+|U| <= |G|+2.

Abstract

In 2003 COHN and UMANS introduced a group-theoretic approach to fast matrix multiplication. This involves finding large subsets S, T and U of a group G satisfying the Triple Product Property (TPP) as a means to bound the exponent $\omega$ of the matrix multiplication. We show that S, T and U may be be assumed to contain the identity and be otherwise disjoint. We also give a much shorter proof of the upper bound |S|+|T|+|U| <= |G|+2.