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<p>We study central configurations when the set of positions is symmetric. We use a theorem (proved in [<CitationRef CitationID="CR33">33</CitationRef>]) 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.</p><p>As an application, we give a complete description of the existence and which masses are possible for central configurations of two nested regulartetrahedra, 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 [<CitationRef CitationID="CR11">11</CitationRef>, <CitationRef CitationID="CR25">25</CitationRef>, <CitationRef CitationID="CR45">45</CitationRef>] and extendedthem by completing the discussion on thecube case, in the inverse and direct problems.</p>

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Decomposition of Symmetrical Classes of Central Configurations

  • 期刊论文

  • 发表日期：

    2026年6月15日

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  • Marcelo P. Santos,
  • Leon D. da Silva

摘要

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 regulartetrahedra, 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 extendedthem by completing the discussion on thecube case, in the inverse and direct problems.