Matrix multiplication : a group - theoretic approach

Given two n × n matrices A and B we want to compute their product c = A · B. The trivial algorithm runs in time n (this and the next running times are meant up to lower order factors n). In 1967 Strassen improved the running time to ≤ n and in 1990 Coppersmith and Winograd improved it further to ≤ n, which continues to be the best to date. Since the output is of size ≥ n it is clear that the best possible running time one can aspire for is n. Remarkably, it is believed that this is attainable. In this lecture we present a group-theoretic approach to matrix multiplication developed by H. Cohn and C. Umans (2003), later with R. Kleinberg and B. Szegedy (2005). This approach gives a conceptually clean way to get fast algorithms, and also provides specific conjectures that if proven yield the optimal n running time. In what follows we first present some notation, then we cover polynomial multiplication, and lastly we present matrix multiplication.