两类一般矩阵的纯迹代数和混合迹代数的Poincare级数
Poincare series of some pure and mixed trace algebras of two generic matrices
摘要：
我们研究特征为零的域K。对n小于或等于6，给出了两个一般n × n矩阵x和y的GL_n-不变量代数C_{n,2}和GL_n-伴随代数T_{n,2}的Poincare级数.简单分级和双分级的情况都包括在内。n最多为4的情况以前是已知的。当n=5或6时，我们证明了C_{n,2}没有参数的二阶系统.对于代数C_{4,2}和C_{5,2}，我们构造了生成元的极小集，并给出了Specht关于两个复矩阵酉相似定理的一个应用.本文对上述各种庞加莱级数的分子和子提出了五点建议。给出了一些启发式公式和待解决的问题。
英文摘要：
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.
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