The geometry of rank decompositions of matrix multiplication I: 2x2   matrices
  This is the first in a series of papers on rank decompositions of the matrix multiplication tensor. In this paper we: establish general facts about rank decompositions of tensors, describe potential ways to search for new matrix multiplication decompositions, give a geometric proof of the theorem of Burichenko's theorem establishing the symmetry group of Strassen's algorithm, and present two particularly nice subfamilies in the Strassen family of decompositions. 
Luca Chiantini, Christian Ikenmeyer, J.M. Landsberg and Giorgio   Ottaviani
cs.CC math.AG