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Seven Coincidences and Sixty Symmetries
Abstract
Motivated by John Baez's question about seven exceptional coincidences among binomial coefficients, we construct seven exterior-power identities for representations of A5. Their common symmetry comes from simplex edges and icosahedral vertices, axes, and faces, together with characters of stabilizers. We enumerate all ordered pairs of representation isomorphism classes in the prescribed dimensions. Apart from the pair of trivial actions, the permutation pair at $1540$ and the complex monomial pair at $7140$ are unique. At $3003$, Klein's twelve-point divisor gives a noncanonical intertwiner factored through polynomial division, differentiation, and residues.
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BibTeX
@misc{lupercio2026seven,
title = {{Seven Coincidences and Sixty Symmetries}},
author = {Ernesto Lupercio and Evgeniya Shaydurova},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00572},
eprint = {2610.00572},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00572},
}
RIS
TY - UNPB
TI - Seven Coincidences and Sixty Symmetries
AU - Ernesto Lupercio
AU - Evgeniya Shaydurova
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00572
UR - https://arxiv.org/abs/2610.00572
AB - Motivated by John Baez's question about seven exceptional coincidences among binomial coefficients, we construct seven exterior-power identities for representations of A_5. Their common symmetry comes from simplex edges and icosahedral vertices, axes, and faces, together with characters of stabilizers. We enumerate all ordered pairs of representation isomorphism classes in the prescribed dimensions. Apart from the pair of trivial actions, the permutation pair at $1540$ and the complex monomial pair at $7140$ are unique. At $3003$, Klein's twelve-point divisor gives a noncanonical intertwiner factored through polynomial division, differentiation, and residues.
ER -