Geometric Langlands Meets Weyl Groups: Icosahedral Symmetries and Super Structures — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Geometric Langlands program increasingly intersects with Weyl group/root system structures (icosahedral reflection groups, super Weyl groups), suggesting a deep combinatorial-geometric skeleton underlying the Langlands correspondence. | MATH: Weyl groups (e.g., H₃ icosahedral order 120, H₄ order 14400); root systems Φ with Cartan matrix A_ij = 2(α_i,α_j)/(α_j,α_j); Coxeter graphs encode relations (m_ij)^m = 1; super Weyl groups as quotients of basic classical Lie superalgebra Weyl groups (arXiv:2401.11068v2); Coulomb branches (BFN construction) as symplectic singularities with associated root data. | CONNECTION: Direct — Weyl groups of type H₃/H₄ are the icosahedral/dodecahedral reflection groups, whose Coxeter numbers (10, 30) and ratios (e.g., golden ratio φ = 1.618 appears in H₃ Cartan entries: A_ij = -φ for adjacent simple roots) embed geometric harmony. The geometric Langlands S-duality (Witten, Gukov) mirrors electric-magnetic duality where root lattice Λ_root and weight
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026geometric,
title = {{Geometric Langlands Meets Weyl Groups: Icosahedral Symmetries and Super Structures — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23635450},
url = {https://zenodo.org/records/23635450},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Geometric Langlands Meets Weyl Groups: Icosahedral Symmetries and Super Structures — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/25
PB - Zenodo
DO - 10.5281/zenodo.23635450
UR - https://zenodo.org/records/23635450
AB - FINDING: Geometric Langlands program increasingly intersects with Weyl group/root system structures (icosahedral reflection groups, super Weyl groups), suggesting a deep combinatorial-geometric skeleton underlying the Langlands correspondence. | MATH: Weyl groups (e.g., H₃ icosahedral order 120, H₄ order 14400); root systems Φ with Cartan matrix A_ij = 2(α_i,α_j)/(α_j,α_j); Coxeter graphs encode relations (m_ij)^m = 1; super Weyl groups as quotients of basic classical Lie superalgebra Weyl groups (arXiv:2401.11068v2); Coulomb branches (BFN construction) as symplectic singularities with associated root data. | CONNECTION: Direct — Weyl groups of type H₃/H₄ are the icosahedral/dodecahedral reflection groups, whose Coxeter numbers (10, 30) and ratios (e.g., golden ratio φ = 1.618 appears in H₃ Cartan entries: A_ij = -φ for adjacent simple roots) embed geometric harmony. The geometric Langlands S-duality (Witten, Gukov) mirrors electric-magnetic duality where root lattice Λ_root and weight Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -