Defining relations of minimal degree of the trace algebra of 3×3 matrices

The trace algebra Cnd over a field of characteristic 0 is generated by all traces of products of d generic n×n matrices, n,d⩾2. Minimal sets of generators of Cnd are known for n=2 and n=3 for any d as well as for n=4 and n=5 and d=2. The defining relations between the generators are found for n=2 and any d and for n=3, d=2 only. Starting with the generating set of C3d given by Abeasis and Pittaluga in 1989, we have shown that the minimal degree of the set of defining relations of C3d is equal to 7 for any d⩾3. We have determined all relations of minimal degree. For d=3 we have also found the defining relations of degree 8. The proofs are based on methods of representation theory of the general linear group and easy computer calculations with standard functions of Maple.