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

Abstract

We work over a field K of characteristic zero. The Poincaré series for the algebra C n , 2 of GL n -invariants and the algebra T n , 2 of GL n -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 C 4 , 2 and C 5 , 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 M n 2 of pairs of n × n matrices with M n ⊗ K 2 , we extend the action of GL n to GL n × GL 2 . For n ⩽ 5 , we compute the Poincaré series for the polynomial invariants of this action when restricted to the subgroups GL n × SL 2 and GL n × Δ 1 , where Δ 1 is the maximal torus of SL 2 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.