Poincaré series of some pure and mixed trace algebras of two generic matrices

We work over a field K of characteristic zero. The Poincaré series for the algebra Cn,2 of GLn-invariants and the algebra Tn,2 of GLn-concomitants of two generic n×n matrices x and y are computed for n⩽6. Both simply graded and bigraded cases are included. The cases n⩽4 were known previously. For C4,2 and C5,2 we construct a minimal set of generators, and give an application to Specht's theorem on unitary similarity of matrices.By identifying the space of pairs of n×n matrices with Mn⊗K2, we extend the action of GLn to GLn×GL2. For n⩽5, we compute the Poincaré series for the polynomial invariants of this action when restricted to the subgroups GLn×SL2 and GLn×Δ1, where Δ1 is the maximal torus of SL2 consisting of diagonal matrices.Five conjectures are proposed concerning the numerators and denominators of various Poincaré series mentioned above. Some heuristic formulas and open problems are stated.