Central configurations with dihedral symmetry

As an application of the representation theory for the dihedral groups, we study the symmetric central configurations in the $ n $-body problem where $ n $ equal masses are placed at the vertices of a regular $ n $-gon. Since the Hessian matrices at these configurations are typically very large, particularly when $ n $ is large, computations of their eigenvalues present a challenging problem. However, by decomposing the action of the dihedral groups into irreducible representations, we show that the Hessians can be simplified to a block-diagonal matrix with small blocks, of the sizes at most $ 2\times 2 $. This is due to fact that the action of a dihedral group can be represented as a block-diagonal matrix with small irreducible blocks. In the end, the eigenvalues can be explicitly obtained by simply computing eigenvalues of these small block matrices. Specifically, we identify the index of the regular $ n $-gon configuration and demonstrate that there are negative eigenvalues for $ n \geq 6 $. This finding contradicts Palmore's theorem, which asserts that the regular $ n $-gon configuration is a local minimum for $ n \geq 6 $.