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

TL;DR

This paper computes Poincare series for certain invariant and concomitant algebras of two generic matrices for n ≤ 6, introduces new generators, and discusses conjectures and open problems in invariant theory.

Contribution

It extends known results to n=5,6, constructs minimal generators for specific cases, and proposes conjectures on Poincare series numerators and denominators.

Findings

C_{n,2} has no bigraded system of parameters for n=5,6

Constructed minimal generators for C_{4,2} and C_{5,2}

Proposed five conjectures on Poincare series properties

Abstract

We work over a field K of characteristic zero. The Poincare 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 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 C_{n,2} has no bigraded system of parameters. For the algebra 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 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.