More a question than an answer, but anyway:
It's plain that if one wants to study representations up to equivalence, one would like to attach to every matrix an invariant by conjugation. One could then look for polynomial functions on End(V) or GL(V) (V a finite dimensional vector space), which display such an invariance. We are really looking for the fixed orbits of the linear action of GL(V) on P(End(V)) (which incidentally, is itself an infinite dimensional representation of an algebraic group). In fact this is a subalgebra of P(End(V)) and maybe one can determine a set of generators. 
David Savitt's [observation](https://mathoverflow.net/a/2815) could lead to the question: does tr(A^n) suffice to generate our algebra?