Defining relations of the noncommutative trace algebra of two 3×3 matrices

The noncommutative (or mixed) trace algebra Tnd is generated by d generic n×n matrices and by the algebra Cnd generated by all traces of products of generic matrices, n,d⩾2. It is known that over a field of characteristic 0 this algebra is a finitely generated free module over a polynomial subalgebra S of the center Cnd. For n=3 and d=2 we have found explicitly such a subalgebra S and a set of free generators of the S-module T32. We give also a set of defining relations of T32 as an algebra and a Gröbner basis of the corresponding ideal. The proofs are based on easy computer calculations with standard functions of Maple, the explicit presentation of C32 in terms of generators and relations, and methods of representation theory of the general linear group.