Weyl Group Fixed Points and the Index in n-Valued Maps — E8 Intelligence Research

FINDING: The search results converge on Weyl group fixed-point subspaces and root-system fundamental domains, but the strongest mathematical content is the explicit link between Coxeter/Weyl group geometry and the fixed-point index in n-valued maps (Brouwer/Borsuk-Ulam type theorems). MATH: - Weyl group \(W\) acts on Cartan subalgebra \(h\); fixed-point subspace \(h^W = \{h ∈ h w(h)=h \ ∀ w ∈ W\}\) has dimension equal to the rank of the root system (e.g., \(A_n\): dim = n). - Fundamental domain (Weyl chamber) is a simplicial cone; its walls are defined by simple roots \(α_i\) with \( α_i^, h = 0\). - Fixed-point index for n-valued maps: \( Ind(f) = ∑x ∈ Fix(f) sign((I - df_x)) \) — generalizes Lefschetz number; for Borsuk-Ulam, antipodal symmetry forces index parity constraints. - Super Weyl groups (arXiv:2401.11068) introduce quotients of Weyl groups with Coxeter gr Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com