Defining relations of invariants of two 3 \times 3 matrices

Abstract

Over a field of characteristic 0, the algebra of invariants of several $n\times n$ matrices under simultameous conjugation by $GL_n$ is generated by traces of products of generic matrices. Teranishi, 1986, found a minimal system of eleven generators of the algebra of invariants of two $3\times 3$ matrices. Nakamoto, 2002, obtained an explicit, but very complicated, defining relation for a similar system of generators over $\mathbb Z$. In this paper we have found another natural set of eleven generators of this algebra of invariants over a field of characteristic 0 and have given the defining relation with respect to this set. Our defining relation is much simpler than that of Nakamoto. The proof is based on easy computer calculations with standard functions of Maple but the explicit form of the relation has been found with methods of representation theory of general linear groups.