Weyl Group Actions on Hyperplane Arrangements: Symmetry, Enumeration, and Invariants — E8 Intelligence Research

FINDING: Weyl group actions on hyperplane arrangements connect root-system symmetries to enumerative geometry and invariant theory, with Tutte polynomials encoding combinatorial structure of symmetric arrangements. | MATH: Weyl groups \(W\) act on root systems \(\Phi\) via reflections \(s_\alpha(x) = x - 2\frac{\langle x,\alpha\rangle}{\langle\alpha,\alpha\rangle}\alpha\); hyperplane arrangement \(\mathcal{A}\) has characteristic polynomial \(\chi_{\mathcal{A}}(t) = \sum_{H \in \mathcal{A}} t^{\dim V_H}\); Tutte polynomial \(T_{\mathcal{A}}(x,y)\) satisfies deletion-contraction \(T = T_{e} + T_{e'}\) and specializes to chromatic polynomial \(P_G(q) = (-1)^{|V|}q^{c(G)}T_G(1-q,0)\). | CONNECTION: Root systems of Weyl groups \(A_n, B_n, C_n, D_n\) have crystallographic symmetries — their Cartan matrices yield eigenvalues tied to \(4\cos^2(\pi/h)\) where \(h\) is Coxeter number (e.g., \(h=6\) for \(G_2\) gives \(4\cos^2(30^\circ)=3\), \(h=30\) for \(E_8\) gives \(4\cos^2(6^\circ)\approx 3 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com