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Gerald Höhn

Publications and source records attributed to Gerald Höhn.

At least 19 recordsLinked to original sources
arXiv2026

The Conway-Parker algebra and the largest Fischer group

We give a direct, self-contained construction of the three sporadic Fischer groups $\mathrm{Fi}_{24}'$, $\mathrm{Fi}_{23}$, and $\mathrm{Fi}_{22}$ from the $783$-dimensional Conway-Parker algebra. We prove that its distinguished roots define involutory algebra automorphisms whose projective actions generate the full Fischer $3$-transposition group $\mathrm{Fi}_{24}$. Its commutator subgroup gives $\mathrm{Fi}_{24}'$, while $\mathrm{Fi}_{23}$ and $\mathrm{Fi}_{22}$ arise as centralizer quotients associated with one and two commuting transpositions. The root and frame geometry determines the group orders and leads to elementary proofs of simplicity, as well as natural rank-three actions and nonsplit central extensions. The construction uses standard facts about the Golay code, Parker's loop, and $M_{24}$. It does not use the Monster or previously known existence or order results for the Fischer groups. Fischer's classification and later recognition theorems are used only for the final identification.

math.GR↗
arXiv2026

A constructive ATLAS of finite simple groups in Lean

We present a constructive atlas of finite simple groups with proofs of their orders, simplicity, and structural properties. The eight completed families are cyclic groups of prime order, alternating groups, the classical series $A_r(q)$, $B_r(q)$, $C_r(q)$, $D_r(q)$, and the exceptional series $G_2(q)$ and the small Ree groups $ {}^2G_2(3^{2m+1})$, $m\geq1$. The fifteen sporadic entries are $M_{11}$, $M_{12}$, $M_{22}$, $M_{23}$, $M_{24}$, $\mathrm{Co}_1$, $\mathrm{Co}_2$, $\mathrm{Co}_3$, $\mathrm{McL}$, $\mathrm{HS}$, $\mathrm{Suz}$, $J_2$, and $\mathrm{Fi}_{22}$, $\mathrm{Fi}_{23}$, $\mathrm{Fi}_{24}'$. The simple parameter ranges and exceptional cases are stated explicitly. The models arise from codes, lattices, forms, algebras, and finite geometries, including the split octonions and the Conway--Parker algebra. They retain natural actions, stabilizers, central quotients, and comparison maps for subsequent group theory. Several classical comparison isomorphisms relate the models, while involution-class counts distinguish the equal-order orthogonal and symplectic families in odd characteristic and rank at least three. Drawing on classical sources and companion mathematical work, the project develops the construction side of finite simple group theory, not the exhaustiveness proof of the classification. The completed models and stated proofs are formalized in Lean for verification and reuse.

cs.SC↗
arXiv2026

Definite integral Albert algebras: arithmetic and finite geometry

We give a unified treatment of definite Albert algebras over $\mathbb{Z}$, with exceptional Siegel--Weil identities as the main arithmetic tool. Starting from the standard octonion order alone, its first scalar theta coefficient saturates the genus average and forces every remaining class to be rootless. A cubic-ring coefficient of the $G_2$ theta lift then gives the mass of the remaining scale-$2$ framed orders, also recovered from shell geometry. A uniform Niemeier--Golay stabilizer bound saturates this mass, proving that there is exactly one further class and determining both automorphism orders, frame transitivity, and the $2457$ frames on $819$ rank-one elements. The associated generalized hexagon identifies the nonstandard group as ${}^3D_4(2):3$ only after its order has been proved. An independent weighted-theta argument gives the trace-zero root-system alternative $\varnothing$ or $A_2E_8^3$. The construction relates positive masses of framed objects to exceptional Eisenstein coefficients.

math.NT↗
arXiv2026

The Hall-Janko Simple Group and the Icosian Leech Lattice

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.

math.GR↗
arXiv2026

The unique extremal threemodular lattice of rank 26, the generalized hexagon $(2,8)$, and the tight Cayley-plane $5$-design

We reconstruct the generalized hexagon of order $(2,8)$ from an abstract even rank-$26$ lattice of determinant $3$ and minimum $4$, and conversely reconstruct the lattice from the hexagon. Harmonic theta identities determine the $819$ shortest vectors in a non-zero discriminant class and their association scheme, and show that every non-empty positive-norm shell of the lattice and its dual is a spherical $5$-design. In the converse direction, the rank-$26$ idempotent gives the lattice, with saturation proved by a short dual-coset argument. A positive-definite Niemeier construction proves existence and uniqueness of the lattice and hence of the hexagon. The third moment of a projective $3$-design in the Cayley plane reconstructs the traceless Albert product. This upgrades angle-preserving bijections to elements of $F_4(\mathbb R)$ and proves geometric uniqueness of the tight $819$-point projective $5$-design. Finally, line deletion gives a rootless index-four sublattice of $N(A_1^{24})$ with a Golay trio and an oriented gluing. The gluing data form eight root-sign orbits, on which $L_3(2)\cong L_2(7)$ acts as on $\mathbf P^1(\mathbf{F}_7)$; two explicit Golay permutations prove transitivity. This proves uniqueness and gives the automorphism-group order. The group is identified afterward as ${}^3D_4(2):3$, with a central factor $C_2$ for the full lattice group. The reconstruction and deletion arguments parallel the length-$26$ binary code construction.

math.CO↗
arXiv2026

Automorphisms of a $(24_4, 32_3)$-configuration on the Schur quartic

We answer the automorphism question raised by Naskręcki and Pokora for their $(24_4, 32_3)$-configuration on the Schur quartic. The stabilizer of either $24$-line half is isomorphic to $W(D_4)\rtimes C_3$, where $C_3$ acts by even triality; the full projective automorphism group has order $1152$. The $D_4$ model also gives an intrinsic coloring and shortens the incidence check.

math.AG↗
arXiv2026

The unique self-dual binary code of length 26 with minimum weight 6

The binary Type~I self-dual [26,13,6] code is classical. We give what appears to be the first direct non-computational proof of its uniqueness. We first determine the weight enumerators of the code and its shadow. Degree-one harmonic MacWilliams identities supply the required 1-designs. Elementary intersection counts then show that the two minimal half-shadows each contain 13 words; these words label the 26 coordinates as 13 points and 13 lines, and the two shadow classes become the point-stars and line-stars. From this structure we give two uniqueness proofs: one reconstructs the projective plane of order 3 and the plane code, including the full automorphism group PGL(3,3):2; the other deletes an intrinsic flag, obtains the odd Golay code together with a deep hole coset, and reconstructs the length 26 code and the size of the automorphism group from this coset datum. Thus the natural length-24 object behind the code is the odd Golay code together with its unique orbit of deep hole cosets.

math.CO↗
arXiv2026

The Kernel of the $\hat A$-Genus in Rational Spin Bordism is Generated by Ricci-Positive Manifolds

We prove that, in every degree, the rational Spin bordism classes represented by manifolds admitting metrics with positive Ricci curvature span exactly the kernel of the $\hat A$-genus. More precisely, for \[ R=Ω_*^{Spin}\otimes\mathbb{Q},\qquad J=\ker(\hat A:R\longrightarrow\mathbb{Q}[u]),\] the $\mathbb{Q}$-span of bordism classes of Ricci-positive Spin manifolds equals $J$ in each degree. This answers, in the differentiable rational Spin category, a question about rational bordism obstructions to positive Ricci curvature which was raised in the context of complex elliptic genera. The proof uses smooth complete intersections of an odd number $\ell$ of quadrics \[ Y_{m,\ell}\subset \mathbb{CP}^{2m+\ell}, \qquad \ell=1,\, 3,\, \ldots,\, 2m-1. \] These manifolds have real dimension $4m$, are Spin and Fano, and therefore admit metrics with positive Ricci curvature. A first-order thickening of the $\hat A$-genus induces $m-1$ linear functionals on $(J/J^2)_{4m}$. Their values on the classes $[Y_{m,\ell}]$ are governed by polynomials $P_{m,q}(\ell)$ of strictly increasing degrees $q+1=1$, $2$, $\ldots$, $m-1$. This gives full rank by a polynomial-interpolation argument.

math.AT↗
arXiv2026

Systematic Orbifold Constructions of Schellekens' Vertex Operator Algebras from Niemeier Lattices

We present a systematic, rigorous construction of all 70 strongly rational, holomorphic vertex operator algebras $V$ of central charge 24 with non-zero weight-one space $V_1$ as cyclic orbifold constructions associated with the 24 Niemeier lattice vertex operator algebras $V_N$ and certain 226 short automorphisms in $\operatorname{Aut}(V_N)$. We show that up to algebraic conjugacy these automorphisms are exactly the generalised deep holes, as introduced in arXiv:1910.04947, of the Niemeier lattice vertex operator algebras with the additional property that their orders are equal to those of the corresponding outer automorphisms. Together with the constructions in arXiv:1708.05990 and arXiv:1910.04947 this gives three different uniform constructions of these vertex operator algebras, which are related through 11 algebraic conjugacy classes in $\operatorname{Co}_0$. Finally, by considering the inverse orbifold constructions associated with the 226 short automorphisms, we give the first systematic proof of the result that each strongly rational, holomorphic vertex operator algebra $V$ of central charge 24 with non-zero weight-one space $V_1$ is uniquely determined by the Lie algebra structure of $V_1$.

math.QA↗
arXiv2025

The Order of the Monster Finite Simple Group

We determine the order of the largest of the twenty-six sporadic simple groups known as the Monster, using a straightforward computational approach. The Monster is here defined as a subgroup of the symmetry group of the 196884-dimensional Griess algebra generated by a group of type $2^{1+24}_+.Co_1$ and an additional triality automorphism. Our approach is based on counting arguments for certain idempotents of the Griess algebra called axes. Our proof is self-contained, requiring only established properties of the Conway group as the automorphism group of the Leech lattice, and some of its subgroups. Although our approach is conceptually simple, it requires extensive calculation inside a 196884-dimensional matrix group that current computer algebra systems cannot easily handle directly. Instead, we use the software package mmgroup, developed by the second author, which supports fast calculations inside the Monster. To our knowledge, this paper contains the first self-contained computation of the order of the Monster. The Monster also acts on the Moonshine module V^#, which is a vertex operator algebra of central charge c=24. We provide a new proof that the Monster is the full automorphism group of the Griess algebra and of the Moonshine module using Borcherds' proof of the Monstrous Moonshine conjectures. In addition, we show that the Monster has exactly two conjugacy classes of involutions. The order of the Baby Monster, the second largest of the sporadic simple groups, is also determined.

math.GR↗
arXiv2024

Classification of Self-Dual Vertex Operator Superalgebras of Central Charge at Most 24

We classify the self-dual (or holomorphic) vertex operator superalgebras of central charge 24, or in physics parlance the purely left-moving, fermionic 2-dimensional conformal field theories with just one primary field. There are exactly 969 such vertex operator superalgebras under suitable regularity assumptions (essentially strong rationality) and the assumption that the shorter moonshine module $V\!B^\natural$ is the unique self-dual vertex operator superalgebra of central charge 23.5 whose weight-1/2 and weight-1 spaces vanish. Additionally, there might be self-dual vertex operator superalgebras arising as fake copies of $V\!B^\natural$ tensored with a free fermion $F$. We construct and classify the self-dual vertex operator superalgebras by determining the 2-neighbourhood graph of the self-dual vertex operator algebras of central charge 24 and also by realising them as simple-current extensions of a dual pair containing a certain maximal lattice vertex operator algebra. We show that all vertex operator superalgebras besides $V\!B^\natural\otimes F$ and potential fake copies thereof stem from elements of the Conway group $\mathrm{Co}_0$, the automorphism group of the Leech lattice $Λ$. By splitting off free fermions $F$, if possible, we obtain the classification for all central charges less than or equal to 24.

math.QA↗
arXiv2019

The Conformal Packing Problem

We formulate the conformal packing problem and dual packing problem in analogy to similar problems for binary codes and lattices. We obtain explicit numerical upper bounds for the minimal dual conformal weight of a unitary strongly-rational vertex operator algebra for several central charges c and we also discuss asymptotic bounds. As a main result, we find the bounds 1 and 2 for the minimal dual conformal weight for the central charges c=8 and c=24, respectively. These bounds are reached by the vertex operator algebra associated to the affine Kac-Moody algebra E_8 at level 1 of central charge c=8 and the moonshine module of central charge c=24. The optimal bounds for these two central charges are obtained by methods similar to the one used by Viazovska and Cohn et al. in solving the sphere packing problem in dimension 8 and 24, respectively.

math-ph↗
arXiv2018

Most vertex superalgebras associated to an odd unimodular lattice of rank 24 have an N=4 superconformal structure

Odd, positive-definite, integral, unimodular lattices N of rank 24 were classified by Borcherds. There are 273 isometry classes of such lattices. Associated to them are vertex superalgebras $V_N$ of central charge c=24. We show that at least 267 of these vertex operator superalgebras contain an N=4 superconformal subalgebra of central charge $c'=6$. This is achieved by studying embeddings $L+\subseteq N$ of a certain rank 6 lattice L+.

math.QA↗
arXiv2018

Finite groups of symplectic automorphisms of hyperkähler manifolds of type $K3^{[2]}$

We determine the possible finite groups $G$ of symplectic automorphisms of hyperkähler manifolds which are deformation equivalent to the second Hilbert scheme of a K3 surface. We prove that $G$ has such an action if, and only if, it is isomorphic to a subgroup of either the Mathieu group $M_{23}$ having at least four orbits in its natural permutation representation on $24$ elements, or one of two groups $3^{1+4}{:}2.2^2$ and $3^4{:}A_6$ associated to $\mathcal{S}$-lattices in the Leech lattice. We describe in detail those $G$ which are maximal with respect to these properties, and (in most cases) we determine all deformation equivalence classes of such group actions. We also compare our results with the predictions of Mathieu Moonshine.

math.AG↗
arXiv2017

On the Genus of the Moonshine Module

We provide a novel and simple description of Schellekens' seventy-one affine Kac-Moody structures of self-dual vertex operator algebras of central charge 24 by utilizing cyclic subgroups of the glue codes of the Niemeier lattices with roots. We also discuss a possible uniform construction procedure of the self-dual vertex operator algebras of central charge 24 starting from the Leech lattice. This also allows us to consider the uniqueness question for all non-trivial affine Kac-Moody structures. We finally discuss our description from a Lorentzian viewpoint.

math.QA↗
arXiv2013

The McKay-Thompson series of Mathieu Moonshine modulo two

In this note, we describe the parity of the coefficients of the McKay-Thompson series of Mathieu moonshine. As an application, we prove a conjecture of Cheng, Duncan and Harvey stated in connection with Umbral moonshine for the case of Mathieu moonshine.

math.NT↗
arXiv2010

The Group of Symmetries of the shorter Moonshine Module

It is shown that the automorphism group of the shorter Moonshine module constructed in my Ph.D. thesis (also called Baby Monster vertex operator superalgebra) is the direct product of the finite simple group known as the Baby Monster and the cyclic group of order 2.

math.QA↗