CENTRAL CONFIGURATIONS WITH DIHEDRAL SYMMETRY

The paper proves that for the regular n-gon central configuration the Hessian of f = sqrt(IU) (and of U) is D_n-invariant, can be block-diagonalized via the irreducible decomposition of the dihedral representation into 1D and 2D pieces, and yields explicit closed-form formulas for the 1D scalars and for the 2×2 blocks whose eigenvalues give the full spectrum. It constructs an orthogonal basis from intersections of eigenspaces of D(s) and (complex) eigenspaces of D(r), derives identities among second partials from invariance, and computes the entries with the characteristic (n−p) multiplicity factor; the 2×2 blocks come in equal pairs Ak = Bk and their eigenvalues are obtained by the quadratic formula. All of these steps appear in Sections 2–3 and Appendix A of the paper, including the character-theoretic decomposition, the invariance identities, explicit block formulas, the equality Bk = Ak, and the spectrum extraction. The candidate solution follows the same representation-theoretic approach, adds a co-rotating/block-circulant viewpoint to motivate the same cosine–sine projections, reproduces the same block templates and eigenvalue formulas, and likewise notes Bk = Ak. One minor caveat: the candidate attributes Bk = Ak to Schur’s lemma alone on the isotypic components, which is stronger than what Schur’s lemma implies in general; in the paper, Bk = Ak is obtained by explicit computation under the stated symmetry (rather than purely abstract multiplicity arguments). Aside from this overreach in justification, the approaches align and the results coincide. See the abstract and Section 2 for the D_n decomposition and invariance, Section 3.1 for the explicit formulas (including the (n−p) factor and Bk = Ak), and the eigenvalue extraction via 2×2 blocks.