标题： 对称性类的中心构型分解 显示英文标题

标题： Decomposition of Symmetrical Classes of Central Configurations

摘要： 我们研究当位置集合具有对称性时的中心配置。 我们使用有限群表示论中的一个定理来探讨中心配置方程的对称性质。 这种方法通过考虑任意数量的物体、对称群和维度，简化了中心配置的方程。 我们讨论如何利用这个定理获得比之前更精细的方程分解。 这里提出的分解方法使用了适应对称性的基方法。 作为应用，我们给出了两个嵌套正四面体、两个嵌套正八面体以及两个嵌套正立方体的中心配置的存在性及可能质量的完整描述。 为此，我们采用了一些有理参数化方法和多元多项式零点的隔离方法。 得到的分解允许使用符号计算来研究这些表达式。 通过这种方式，我们总结了之前的讨论，并通过完成对立方体情况的分析，扩展了这些讨论，包括逆问题和正问题。 显示英文摘要

摘要： We study central configurations when the set of positions is symmetric. We use a theorem from representation theory of finite groups to explore the symmetry properties of equations for central configurations. This approach simplifies equations for central configurations by considering arbitrary numbers of bodies, symmetry groups, and dimensions. We discuss how to use this theorem to obtain a more refined decomposition of the equations than that given before. The decomposition presented here uses the symmetry-adapted basis method. As an application, we give a complete description of the existence and which masses are possible for central configurations of two nested regular tetrahedrons, two nested regular octahedrons, and two nested regular cubes. To do this, we employ some methods of rational parameterizations and isolation of zeros of multivariate polynomials. The decomposition obtained allows symbolic calculations to be used to study the expressions. This way, we summarize previous discussions and extend them by completing the analysis on the cube case, for both the inverse and direct problems.