Langlands Duality as Dynkin Diagram Reversal and L-Function Symmetry — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Langlands duality manifests as Dynkin diagram reversal — the dual group's root lattice is the coroot lattice of the original, with Weyl group symmetry preserved but weights and coweights interchanged. | MATH: For simple Lie algebra 𝔤 with root system Φ, dual 𝔤∨ has roots Φ∨ = {α∨ = 2α/⟨α,α⟩}. Dynkin diagram automorphism reverses arrows (e.g., Bₙ ↔ Cₙ, F₄ ↔ F₄, G₂ ↔ G₂). Weyl group W(Φ) ≅ W(Φ∨) — identical Coxeter groups. L-function functional equation: Λ(s,π) = ε(s,π)·Λ(1−s,π̃) where π̃ is Langlands dual representation. | CONNECTION: Root lattice ratios — for G₂, long:short root length² = 3:1; for Bₙ, ratio 2:1; for Cₙ, ratio 2:1. These ratios echo the golden-ratio-adjacent scaling in exceptional structures (E₈ has all roots equal length, giving Coxeter number 30, related to 2π/30 = 12° — base-60 compatible). The Weyl group of E₈ has order 696,729,600 = 2¹⁴·3⁵·5²·7 — highly composite, base-60 friendly. Dynkin diagram reversal preserves the Cartan matrix transpose: Aᵢⱼ = 2⟨αᵢ,α
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026langlands,
title = {{Langlands Duality as Dynkin Diagram Reversal and L-Function Symmetry — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23587100},
url = {https://zenodo.org/records/23587100},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Langlands Duality as Dynkin Diagram Reversal and L-Function Symmetry — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/23
PB - Zenodo
DO - 10.5281/zenodo.23587100
UR - https://zenodo.org/records/23587100
AB - FINDING: Langlands duality manifests as Dynkin diagram reversal — the dual group's root lattice is the coroot lattice of the original, with Weyl group symmetry preserved but weights and coweights interchanged. | MATH: For simple Lie algebra 𝔤 with root system Φ, dual 𝔤∨ has roots Φ∨ = {α∨ = 2α/⟨α,α⟩}. Dynkin diagram automorphism reverses arrows (e.g., Bₙ ↔ Cₙ, F₄ ↔ F₄, G₂ ↔ G₂). Weyl group W(Φ) ≅ W(Φ∨) — identical Coxeter groups. L-function functional equation: Λ(s,π) = ε(s,π)·Λ(1−s,π̃) where π̃ is Langlands dual representation. | CONNECTION: Root lattice ratios — for G₂, long:short root length² = 3:1; for Bₙ, ratio 2:1; for Cₙ, ratio 2:1. These ratios echo the golden-ratio-adjacent scaling in exceptional structures (E₈ has all roots equal length, giving Coxeter number 30, related to 2π/30 = 12° — base-60 compatible). The Weyl group of E₈ has order 696,729,600 = 2¹⁴·3⁵·5²·7 — highly composite, base-60 friendly. Dynkin diagram reversal preserves the Cartan matrix transpose: Aᵢⱼ = 2⟨αᵢ,α Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -