Plethysm and fast matrix multiplication

TL;DR

This paper explores the plethysm of the adjoint representation of SL_n, decomposes it for k=3, and relates findings to fast matrix multiplication techniques like the Coppersmith-Winograd tensor.

Contribution

It provides the first detailed decomposition of $S^3(rak{sl}_n)$ and identifies highest weight vectors, connecting representation theory to efficient matrix multiplication algorithms.

Findings

Decomposition of $S^3(rak{sl}_n)$ into irreducibles

Identification of highest weight vectors for all components

Connections established between plethysm and fast matrix multiplication

Abstract

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 $SL_n$. 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.