Weyl Orbits and Cartan Transpose Symmetries in Kac-Moody Root Systems — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Weyl group orbit structure on Kac-Moody root systems is governed by Cartan matrix transpose symmetries, linking long/short root ratios to Langlands dual data. MATH: For a root system with Cartan matrix Aij = 2(αi,αj)/(αj,αj), the dual system uses AT. Long/short root length ratio squared = 2 for simply-laced (all roots equal), but for Bn: long²/short² = 2; for Cn: ratio = 2 (dual swap); for G2: long²/short² = 3. Weyl group W acts on Δ, with orbit structure determined by W-invariant bilinear form on h*. The Kac-Moody generalization preserves the orbit count via AT — e.g., for affine A1(1), the imaginary root δ has W-orbit size 1, while real roots split into long/short orbits with multiplicities tied to AT entries. CONNECTION: Root length ratios {1, √2, √3} map to crystallographic constraints. The ratio √2 appears in Bn/Cn — related to 0.707 (ha
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026weyl,
title = {{Weyl Orbits and Cartan Transpose Symmetries in Kac-Moody Root Systems — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23553857},
url = {https://zenodo.org/records/23553857},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Weyl Orbits and Cartan Transpose Symmetries in Kac-Moody Root Systems — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/22
PB - Zenodo
DO - 10.5281/zenodo.23553857
UR - https://zenodo.org/records/23553857
AB - FINDING: Weyl group orbit structure on Kac-Moody root systems is governed by Cartan matrix transpose symmetries, linking long/short root ratios to Langlands dual data. MATH: For a root system with Cartan matrix A_ij = 2(α_i,α_j)/(α_j,α_j), the dual system uses A^T. Long/short root length ratio squared = 2 for simply-laced (all roots equal), but for B_n: long²/short² = 2; for C_n: ratio = 2 (dual swap); for G_2: long²/short² = 3. Weyl group W acts on Δ, with orbit structure determined by W-invariant bilinear form on h^*. The Kac-Moody generalization preserves the orbit count via A^T — e.g., for affine A_1^(1), the imaginary root δ has W-orbit size 1, while real roots split into long/short orbits with multiplicities tied to A^T entries. CONNECTION: Root length ratios {1, √2, √3} map to crystallographic constraints. The ratio √2 appears in B_n/C_n — related to 0.707 (ha Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -