Decomposition of Symmetrical Classes of Central Configurations

TL;DR

This paper applies representation theory to decompose equations for symmetric central configurations, enabling a detailed analysis of configurations like nested regular polyhedra and extending previous results with symbolic computation methods.

Contribution

It introduces a symmetry-adapted basis method for decomposing equations of central configurations, providing a comprehensive framework for analyzing symmetric arrangements of bodies.

Findings

Complete description of configurations for nested regular polyhedra

Identification of possible masses for symmetric configurations

Extension of analysis to cube configurations

Abstract

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.