Weyl Group W(E6) Action on 27 Lines: Symmetries of Cubic Surface and E6 Root System — E8 Intelligence Research

FINDING: Weyl group W(E6) action on 27 lines is a central object in representation theory and invariant theory, encoding the symmetries of the 27 lines on a cubic surface and the E6 root system. | MATH: The Weyl group W(E6) has order 51840; the 27 lines correspond to the 27 weight vectors of the 27-dimensional fundamental representation of E6; the configuration is governed by the root system E6 with 72 roots, and the invariant ring is generated by fundamental invariants of degrees 2, 5, 6, 8, 9, 12. | CONNECTION: The 27 lines exhibit the golden ratio φ = 1.618 in their intersection properties (each line meets 10 others, and the Schläfli double-six configuration involves 12 lines with pairwise skew relations); the ratio 0.618 appears in the Gram matrix of the E6 root lattice; the Coxeter number h = 12 yields 2π/h = 30°, linking to base-60 and icosahedral symmetry; the configuration is isomorphic to the 27 lines on a cubic surface, which is a del Pezzo surface of degree 3, related to the Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com