Transpose of Cartan Matrices and Langlands Dual Root Systems — E8 Intelligence Research

FINDING: The search results are pedagogical (YouTube lectures) rather than novel research; they collectively point to the structural role of the transpose in Cartan matrices and root-length inversion, but contain no new theorem or derived constant. | MATH: For an irreducible root system, the Cartan matrix \\(A_{ij} = 2\\langle \\alpha_i, \\alpha_j\\rangle/\\langle \\alpha_j, \\alpha_j\\rangle\\). The transpose \\(A^T\\) corresponds to the dual root system \\(\\check{\\Phi}\\) (Langlands dual), where long roots become short and vice versa. Length ratio squared: for simply-laced (A,D,E) all roots equal, ratio = 1; for non-simply-laced (B,C,F,G), the squared length ratio of long:short is 2:1 (B,C,F) or 3:1 (G). The inverse transpose \\(A^{-T}\\) maps the weight lattice to the coweight lattice. | CONNECTION: The dual root system preserves the Weyl group (same symmetry), but swaps the fundamental chamber's wall lengths — this is a crystallographic duality. The ratio 2:1 and 3:1 are the only possible non-triv Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com