紧线性群轨道空间中地层的有理参数化
Rational parametrization of strata in orbit spaces of compact linear groups
摘要：
在欧氏空间$\real^n$中作用的紧致线性群$G$的变换下协变或不变的函数可以作为定义在群的轨道空间中的函数来研究。轨道空间是由具有相同对称性的$G$ -轨道构成的半代数流形的有限层集合的并集。在本文中，我们提供了一个简单的配方，以获得合理的参数化的地层。我们的结果可以很容易地利用，在许多物理背景下，协变或不变的功能的研究是重要的，例如在确定的模式的自发对称性破缺，在分析相空间和结构相变（朗道理论），在协变分岔理论，在晶体场论和在大多数领域的固态理论中使用的对称性适应的功能。最后给出了该配方的应用实例。
英文摘要：
Functions which are covariant or invariant under the transformations of a compact linear group $G$ acting in a euclidean space $\real^n$, can be profitably studied as functions defined in the orbit space of the group. The orbit space is the union of a finite set of strata, which are semialgebraic manifolds formed by the $G$-orbits with the same symmetry. In this paper we provide a simple recipe to obtain rational parametrizations of the strata. Our results can be easily exploited, in many physical contexts where the study of covariant or invariant functions is important, for instance in the determination of patterns of spontaneous symmetry breaking, in the analysis of phase spaces and structural phase transitions (Landau's theory), in covariant bifurcation theory, in crystal field theory and in most areas of solid state theory where use is made of symmetry adapted functions. An example of utilization of the recipe is also discussed at the end of the paper.
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