A rank 18 Waring decomposition of $sM_{\langle 3\rangle}$ with 432 symmetries

TL;DR

This paper provides an explicit rank 18 Waring decomposition of the symmetrized matrix multiplication tensor for 3x3 matrices, revealing its symmetry group and advancing understanding of matrix multiplication complexity.

Contribution

It introduces a new explicit decomposition of the tensor with rank 18 and analyzes its symmetry group, contributing to the study of matrix multiplication complexity.

Findings

Explicit rank 18 Waring decomposition of $sM_{<3>}$

Identification of the symmetry group of the decomposition

Insights into the tensor's structure and symmetries

Abstract

The recent discovery that the exponent of matrix multiplication is determined by the rank of the symmetrized matrix multiplication tensor has invigorated interest in better understanding symmetrized matrix multiplication. I present an explicit rank 18 Waring decomposition of $sM_{\langle 3\rangle}$ and describe its symmetry group.