The Simultaneous Triple Product Property and Group-theoretic Results for the Exponent of Matrix Multiplication

This review paper describes certain elementary methods of group theory for studying the algebraic complexity of matrix multiplication, as measured by the exponent 2 ≤ ω ≤ 3 of matrix multiplication. which is conjectured to be 2. The seed of these methods lies in two ideas of H. Cohn, C. Umans et. al., firstly, that it is possible to " realize " a matrix product via the regular algebra of a (finite) group having a triple of subsets satisfying the so-called triple product property (TPP), and more generally, that it is possible to simultaneously realize several independent matrix products via a single group having a family of triples of subsets satisfying the so-called simultaneous triple product property (STPP), in such a way that the complexity of these several multiplications does not exceed the complexity of one multiplication in the regular algebra of the group. The STPP, in particular, has certain implications for ω which we describe.