Plethysm and fast matrix multiplication

Motivated by the symmetric version of matrix multiplication we study the plethysm $S^k({\mathfrak{sl}}_n)$ of the adjoint representation ${\mathfrak{sl}}_n$ 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.