Matrix multiplication is an indispensable operation in mathematics with applications in computer science, engineering, and physics. Researchers H. Cohn and C. Umans proposed a framework for fast matrix multiplication algorithms relying on two steps: mapping matrices to group rings and applying a result in mathematics known as the Wedderburn-Artin Theorem. The former step is well-understood and has been implemented for certain cases by M. Anderson; we focus on the latter step.
In particular, the Wedderburn-Artin Theorem guarantees a mapping between group rings and direct sums of matrix algebras over division rings. We follow G. Ivanyos, K. Friedl, L. Rònyai, and others whose combined works form an algorithm for the computation of this mapping for elements of the group ring. We produce an implementation of their construction. We demonstrate success with group rings over the rationals that map to matrix algebras of dimension 1 over the rationals. We show progress made towards a more general implementation of the computation of these mappings, and thus towards the H. Cohn and C. Umans fast matrix multiplication algorithms.