Defining relation for semi-invariants of three by three matrix triples

TL;DR

This paper explicitly computes the fundamental relation defining the algebra of invariants for triples of 3x3 matrices under the action of SL_3×SL_3, linking it to other key invariant algebras.

Contribution

It provides the first explicit description of the defining relation for the algebra of invariants of three 3x3 matrices under the specified group action.

Findings

Explicit defining relation for the algebra of invariants is obtained.

Connections to other prominent invariant algebras are established.

The result advances understanding of matrix invariants under group actions.

Abstract

The single defining relation of the algebra of $SL_3\times SL_3$-invariants of triples of $3\times 3$ matrices is explicitly computed. Connections to some other prominent algebras of invariants are pointed out.