The Hall-Janko Simple Group and the Icosian Leech Lattice
Abstract
We give a unified arithmetic and geometric treatment of the Hall-Janko simple group and the icosian Leech lattice. Let ℐ be the standard icosian maximal order over K=ℚ(√5). Following the mass-formula and Venkov framework for the Niemeier and Leech lattices, we classify positive definite unimodular Hermitian ℐ-lattices of quaternionic rank $3$. Hashimoto's mass formula, the Siegel-Weil average, and Gundlach's structure theorem determine the scalar theta series of every nonsplit class. A two-place harmonic separation of the Leech minimal shell proves that its three Hermitian components are quaternionic projective $5$-designs. For the norm-$2$ shell, integrality and projective moments recover the complete angle scheme and its $525$ orthogonal frames. The finite code of one frame then saturates the residual mass, giving exactly the split and Tits classes and an independent computation of the order of Autℐ(LT)≅2.J2. Finally, a quaternionic line system with the prescribed five projective inner products has at most $315$ points, with equality precisely for the Hall-Janko configuration up to PSp(3). Equality in Hoggar's bound supplies the design property; Cohen-Tits uniqueness, golden-angle fission, and the centered cubic moment give geometric rigidity.
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BibTeX
@misc{hohn2026halljanko,
title = {{The Hall-Janko Simple Group and the Icosian Leech Lattice}},
author = {Gerald Höhn},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22842},
eprint = {2609.22842},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22842},
}
RIS
TY - UNPB
TI - The Hall-Janko Simple Group and the Icosian Leech Lattice
AU - Gerald Höhn
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.22842
UR - https://arxiv.org/abs/2609.22842
AB - We give a unified arithmetic and geometric treatment of the Hall-Janko simple group and the icosian Leech lattice. Let ℐ be the standard icosian maximal order over K=ℚ(√5). Following the mass-formula and Venkov framework for the Niemeier and Leech lattices, we classify positive definite unimodular Hermitian ℐ-lattices of quaternionic rank $3$. Hashimoto's mass formula, the Siegel-Weil average, and Gundlach's structure theorem determine the scalar theta series of every nonsplit class. A two-place harmonic separation of the Leech minimal shell proves that its three Hermitian components are quaternionic projective $5$-designs. For the norm-$2$ shell, integrality and projective moments recover the complete angle scheme and its $525$ orthogonal frames. The finite code of one frame then saturates the residual mass, giving exactly the split and Tits classes and an independent computation of the order of Aut_ℐ(L_T)≅2.J_2. Finally, a quaternionic line system with the prescribed five projective inner products has at most $315$ points, with equality precisely for the Hall-Janko configuration up to PSp(3). Equality in Hoggar's bound supplies the design property; Cohen-Tits uniqueness, golden-angle fission, and the centered cubic moment give geometric rigidity.
ER -