The Hall-Janko Simple Group and the Icosian Leech Lattice

Source abstract

We give a unified arithmetic and geometric treatment of the Hall-Janko simple group and the icosian Leech lattice. Let $\mathcal{I}$ be the standard icosian maximal order over $K=\mathbb{Q}(\sqrt5)$. Following the mass-formula and Venkov framework for the Niemeier and Leech lattices, we classify positive definite unimodular Hermitian $\mathcal{I}$-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 $\operatorname{Aut}_{\mathcal{I}}(L_{\mathrm T})\cong2.J_2$. 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 $P\operatorname{Sp}(3)$. Equality in Hoggar's bound supplies the design property; Cohen-Tits uniqueness, golden-angle fission, and the centered cubic moment give geometric rigidity.