McKay Correspondence and Monstrous Moonshine: Unifying Finite Groups and Modular Forms — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: The McKay correspondence links finite subgroups of SU(2) (and SL(2,C)) to affine ADE Dynkin diagrams, while Monstrous Moonshine connects the Monster group's irreducible representation dimensions to the Fourier coefficients of the modular j-invariant. | MATH: - **McKay correspondence (classical):** For a finite subgroup G ⊂ SU(2), the McKay graph is an affine ADE Dynkin diagram. The vertices correspond to irreducible representations ρi of G, and edges are given by the tensor product decomposition: ρi ⊗ ℂ2 = ⨁j aij ρj, where ℂ2 is the defining 2D representation. The Cartan matrix of the affine Dynkin diagram is C = 2I − A, where A = (aij). - **Monstrous Moonshine (McKay, 1978):** The first nontrivial irreducible representation of the Monster group M has dimension 196,883. The j-invariant expansion: \( j(\tau) = q^{-1} + 744 + 196,884 q + 21,493,760 q^2 + 864,299,970 q^3 + \
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026mckay,
title = {{McKay Correspondence and Monstrous Moonshine: Unifying Finite Groups and Modular Forms — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23554120},
url = {https://zenodo.org/records/23554120},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - McKay Correspondence and Monstrous Moonshine: Unifying Finite Groups and Modular Forms — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/22
PB - Zenodo
DO - 10.5281/zenodo.23554120
UR - https://zenodo.org/records/23554120
AB - FINDING: The McKay correspondence links finite subgroups of SU(2) (and SL(2,C)) to affine ADE Dynkin diagrams, while Monstrous Moonshine connects the Monster group's irreducible representation dimensions to the Fourier coefficients of the modular j-invariant. | MATH: - **McKay correspondence (classical):** For a finite subgroup G ⊂ SU(2), the McKay graph is an affine ADE Dynkin diagram. The vertices correspond to irreducible representations ρ_i of G, and edges are given by the tensor product decomposition: ρ_i ⊗ ℂ^2 = ⨁_j a_ij ρ_j, where ℂ^2 is the defining 2D representation. The Cartan matrix of the affine Dynkin diagram is C = 2I − A, where A = (a_ij). - **Monstrous Moonshine (McKay, 1978):** The first nontrivial irreducible representation of the Monster group M has dimension 196,883. The j-invariant expansion: \( j(\tau) = q^{-1} + 744 + 196,884 q + 21,493,760 q^2 + 864,299,970 q^3 + \ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -