Theory of invariants and applications

我们将分析经典群G的多项式不变量GL_n， SO_n， O_n和Sp_{2n}作用于适当大小（n或2n）的矩阵对（X，Y）的代数结构。特别地，我们的目的是计算（对于一些较小的n值）庞加莱级数，齐次发生器的最小集，齐次参数系统，相应的Hironaka分解，并描述作用的Hilbert零锥。这种明确的结果仅在少数情况下已知，重要的是要为各种应用程序提供这些数据。例如，这些结果可用于研究满足多项式恒等式的代数。

We shall analyze the structure of the algebra of polynomial invariants of a classical group G such as GL_n, SO_n, O_n and Sp_{2n} acting on pairs of matrices (X,Y) of appropriate size (n or 2n) by simultaneous conjugation. In particular our intention is to compute (for some small values of n) the Poincare series, a minimal set of homogeneous generators, a homogeneous system of parameters, the corresponding Hironaka decomposition and also to describe the Hilbert's null-cone for the action. Such explicit results are known only in a handful of cases and it is important to have these data available for various applications. For instance these results may be used in the study of algebras satisfying polynomial identities.