Upgrading Subgroup Triple Product Property Triples

In 2003 COHN and UMANS introduced a group-theoretic approach to fast matrix multiplication. This involves finding large subsets of a group $G$ satisfying the Triple Product Property (TPP) as a means to bound the exponent $\omega$ of matrix multiplication. Recently, Hedtke and Murthy discussed several methods to find TPP triples. Because the search space for subset triples is too large, it is only possible to focus on subgroup triples. We present methods to upgrade a given TPP triple to a bigger TPP triple. If no upgrade is possible we use reduction methods (based on random experiments and heuristics) to create a smaller TPP triple that can be used as input for the upgrade methods. If we apply the upgrade process for subset triples after one step with the upgrade method for subgroup triples we achieve an enlargement of the triple size of 100 % in the best case.

💡 Research Summary

The paper addresses a central problem in the Cohn‑Umans group‑theoretic framework for fast matrix multiplication: how to find large subsets of a finite group G that satisfy the Triple Product Property (TPP) and thereby improve the known upper bounds on the matrix‑multiplication exponent ω. In the original framework the size of a TPP triple (S,T,U) – measured by |S|·|T|·|U| – directly determines how large a matrix multiplication can be embedded in the group algebra ℂ