  We introduce a relaxation of the notion of tensor rank, called s-rank, and
show that upper bounds on the s-rank of the matrix multiplication tensor imply
upper bounds on the ordinary rank. In particular, if the "s-rank exponent of
matrix multiplication" equals 2, then omega = 2. This connection between the
s-rank exponent and the ordinary exponent enables us to significantly
generalize the group-theoretic approach of Cohn and Umans, from group algebras
to general algebras. Embedding matrix multiplication into general algebra
multiplication yields bounds on s-rank (not ordinary rank) and, prior to this
paper, that had been a barrier to working with general algebras.
  We identify adjacency algebras of coherent configurations as a promising
family of algebras in the generalized framework. Coherent configurations are
combinatorial objects that generalize groups and group actions; adjacency
algebras are the analogue of group algebras and retain many of their important
features. As with groups, coherent configurations support matrix multiplication
when a natural combinatorial condition is satisfied, involving triangles of
points in their underlying geometry.
  Finally, we prove a closure property involving symmetric powers of adjacency
algebras, which enables us to prove nontrivial bounds on omega using
commutative coherent configurations and suggests that commutative coherent
configurations may be sufficient to prove omega = 2. Altogether, our results
show that bounds on omega can be established by embedding large matrix
multiplication instances into small commutative coherent configurations, while
avoiding the representation-theoretic complications that were present in the
group-theoretic approach.
