Poincare series of some pure and mixed trace algebras of two generic matrices
Dragomir Z. Djokovic
Abstract
We work over a field K of characteristic zero. The Poincare series for the algebra Cn,2 of GLn-invariants and the algebra Tn,2 of GLn-concomitants of two generic n x n matrices x and y are presented for n less than or equal 6. Both simply graded and bigraded cases are included. The cases for n at most 4 were known previously. If n=5 or 6, we show that Cn,2 has no bigraded system of parameters. For the algebra C4,2 and C5,2 we construct a minimal set of generators and give an application to Specht's theorem on unitary similarity of two complex matrices. Five conjectures are proposed concerning the numerators and denominators of various Poincare series mentioned above. Some heuristic formulas and open problems are stated.
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