紧共正则线性群轨道空间的确定
The Determination of the Orbit Spaces of Compact Coregular Linear Groups
摘要：
相变的某些方面可以更方便地在对称群作用的轨道空间中进行研究。在简要回顾了这种方法的基本思想之后，我将集中讨论数学方面，更确切地说，是确定轨道空间及其地层的方程。我将只处理紧共正则线性群。所揭示的方法是与G. Sartori和它是基于一个矩阵微分方程的解决方案。这样的方程很容易解决，如果一个完整的基础是已知的。如果完整性基是未知的，那么无论如何都可以确定对于基本不变量的哪些阶有方程的解，并且在所有这些情况下也可以找出解的显式形式。这些解完全决定了轨道空间的分层。这种计算已在二维、三维和四维轨道空间中进行。该方法具有普遍的有效性，但随着轨道空间维数$q$的增加，计算的复杂性大大增加。也发现了一些归纳规则。一旦已知了$q$维情形的解，它们就可以很容易地确定$(q+1)$维情形的大多数解.暴露的方法是有趣的，因为它允许确定轨道空间，而不使用任何特定的知识组结构和完整性的基础和证据的某些隐藏的，但未知的联系与群论和不变理论。
英文摘要：
Some aspects of phase transitions can be more conveniently studied in the orbit space of the action of the symmetry group. After a brief review of the fundamental ideas of this approach, I shall concentrate on the mathematical aspect and more exactly on the determination of the equations defining the orbit space and its strata. I shall deal only with compact coregular linear groups. The method exposed has been worked out together with prof. G. Sartori and it is based on the solution of a matrix differential equation. Such equation is easily solved if an integrity basis of the group is known. If the integrity basis is unknown one may determine anyway for which degrees of the basic invariants there are solutions to the equation, and in all these cases also find out the explicit form of the solutions. The solutions determine completely the stratification of the orbit spaces. Such calculations have been carried out for 2, 3 and 4-dimensonal orbit spaces. The method is of general validity but the complexity of the calculations rises tremendously with the dimension $q$ of the orbit space. Some induction rules have been found as well. They allow to determine easily most of the solutions for the $(q+1)$-dimensional case once the solutions for the $q$-dimensional case are known. The method exposed is interesting because it allows to determine the orbit spaces without using any specific knowledge of group structure and integrity basis and evidences a certain hidden and yet unknown link with group theory and invariant theory.
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