COS 598 D Lecture 10 Applications of Group Representation

We let ω denote the least exponent such that two n× n matrices can be multiplied with O(nω+ ) arithmetic operations for every > 0. It is clear that ω ≥ 2, while Strassen showed that ω is strictly less than 3. Today it is widely believed that ω = 2, although the best upper bound is roughly 2.34 due Coppersmith and Winograd. We will see a somewhat worse upper bound based on a group theoretic approach due to Cohen and Umans.