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 \(\mathfrak{h}\); fixed-point subspace \(\mathfrak{h}^W = \{h \in \mathfrak{h} \mid w(h)=h \ \forall w \in W\}\) 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 \(\alpha_i\) with \(\langle \alpha_i^\vee, h \rangle = 0\). - Fixed-point index for n-valued maps: \( \text{Ind}(f) = \sum_{x \in \text{Fix}(f)} \text{sign}(\det(I - df_x)) \) — 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