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 \(Φ\) via reflections \(s_α(x) = x - 2 x,α/α,αα\); hyperplane arrangement \(A\) has characteristic polynomial \(χA(t) = ∑H ∈ A tV_H\); Tutte polynomial \(TA(x,y)\) satisfies deletion-contraction \(T = Tₑ + Te'\) and specializes to chromatic polynomial \(P_G(q) = (-1)|V|qc(G)T_G(1-q,0)\). | CONNECTION: Root systems of Weyl groups \(A_n, B_n, C_n, D_n\) have crystallographic symmetries — their Cartan matrices yield eigenvalues tied to \(4cos^2(π/h)\) where \(h\) is Coxeter number (e.g., \(h=6\) for \(G_2\) gives \(4cos^2(30^∘)=3\), \(h=30\) for \(E_8\) gives \(4cos^2(6^∘)≈ 3 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com