Decomposition of Symmetrical Classes of Central Configurations

We study central configurations when the set of positions is symmetric. We use a theorem (proved in 33 ) that allows us to use the 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 tetrahedra, 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. In this way, we summarized the same discussions of works done in 11 , 25 , 45 and extended them by completing the discussion on the cube case, in the inverse and direct problems.