Defining Relations of Minimal Degree of the Trace Algebra of $3 \times 3$ Matrices

Abstract

The trace algebra C(n,d) over a field of characteristic 0 is generated by all traces of products of d generic nxn matrices, n,d>1. Minimal sets of generators of C(n,d) 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 C(3,d) given by Abeasis and Pittaluga in 1989, we have shown that the minimal degree of the set of defining relations of C(3,d) is equal to 7 for any d>2. 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.