Plethysm and fast matrix multiplication

Motivated by the symmetric version of matrix multiplication we study the plethysm $S^k({\mathfrak{s}\mathfrak{l}}_n)$ of the adjoint representation ${\mathfrak{s}\mathfrak{l}}_n$ of the Lie group $S{L}_n$. In particular, we describe the decomposition of this representation into irreducible components for $k=3$, and find highest weight vectors for all irreducible components. Relations to fast matrix multiplication, in particular the Coppersmith-Winograd tensor are presented.