Symmetrizing the Matrix Multiplication Tensor

I will consider the symmetrization of the matrix multiplication tensor. This is the cubic polynomial trace(A^3), where A is a n*n matrix. Its interest relies on the fact that the asymptotic exponent for its rank (when n goes to infinity) is the same than the constant omega which measures the complexity of matrix multiplication. On the other hand, a cubic polynomial is more tractable and allows more algebraic geometry tools.