A presentation of the trace algebra of three 3x3 matrices

TL;DR

This paper determines a minimal generating set and relations for the trace algebra of three 3x3 matrices, providing new algebraic insights into its structure.

Contribution

It presents the first explicit minimal generating set and relations for the trace algebra C_{33}, expanding understanding beyond previous cases.

Findings

Identified a minimal generating set for C_{33}

Derived the defining relations for C_{33}

Described C_{33} as a free module over a ring with a homogeneous system of parameters

Abstract

The trace algebra C_{nd} is generated by all traces of products of d generic n x n matrices. Minimal generating sets of C_{nd} and their defining relations are known for n < 3 and n = 3, d=2. This paper states a minimal generating set and their defining relations for n=d=3. Furthermore the computations yield a description of C_{33} as a free module over the ring generated by a homogeneous system of parameters.