Spectral Gap of Coxeter Elements Linked to Cartan Matrices via Coxeter Number — E8 Intelligence Research

FINDING: The spectral gap of a Coxeter element in an irreducible root system is governed by the eigenvalue \(4\cos^2(\pi/h)\), linking Cartan matrices to spectral geometry. | MATH: For a Coxeter element \(c\) with Coxeter number \(h\), the eigenvalues are \(e^{2\pi i k/h}\) (for \(k\) coprime to \(h\) in the simply-laced case). The largest eigenvalue of the Cartan matrix \(A\) (or the adjacency matrix of the associated Dynkin diagram) is \(2\cos(\pi/h)\) for \(A_2\) (triangle), but for general \(A_n\) the spectral radius of the Cartan matrix is \(2\cos(\pi/(n+1))\). The specific value \(4\cos^2(\pi/h)\) appears as the top eigenvalue of the **Coxeter transformation** squared (or the Gram matrix of the root system) — e.g., for \(h=5\) (type \(A_4\)), \(4\cos^2(\pi/5)=4\cdot(0.809)^2=2.618\) (the golden ratio squared). For \(h=6\) (type \(D_4\) or \(B_3\)), \(4\cos^2(\pi/6)=3\). For \(h=10\) (type \(H_4\)), \(4\cos^2(\pi/10)=4\cdot(0.951)^2=3.618 = 2+\sqrt{5}\). | CONNECTION: The golden r Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com