arXiv subjects

Robert L. Griess Jr.

Publications and source records attributed to Robert L. Griess Jr..

10 recordsLinked to original sources

arXiv2026

Lattice vertex algebras over a field of positive characteristic: degenerate cases

A vertex operator algebra $V_L$ associated with a positive definite even lattice $L$ has a standard integral form, which we denote it by $V_{L,\mathbb{Z}} $. If $F$ is a field of characteristic $p>0$, it is known that $V_{L,F}:= F\otimes_\mathbb{Z} V_{L,\mathbb{Z}}$, a vertex algebra over $F$, is simple if and only if $(p, \det(L))=1$. In this article, we study $V_{L,F}$ when the characteristic of $F$ divides $\det(L)$. We determine the radical $\mathrm{Rad}$ of the invariant bilinear form on $V_{L,F}$, show that it is the unique maximal ideal and study the quotient vertex algebra $V_{L,F}/\mathrm{Rad}$.

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arXiv2013

Applications of vertex algebra covering procedures to Chevalley groups and modular moonshine

A vertex operator algebra of lattice type ADE has a standard integral form which extends a Chevalley basis for its degree 1 Lie algebra. This integral form may be used to define a vertex algebra over a commutative ring $R$ and to get a Chevalley group over $R$ of the same type, acting as automorphisms of this vertex algebra. We define vertex algebras of types BCFG over a commutative ring and certain reduced VAs, then get analogous results about automorphism groups. In characteristics 2 and 3, there are exceptionally large automorphism groups. A covering algebra idea of Frohardt and Griess for Lie algebras is applied to the vertex algebra situation. We use integral form and covering procedures for vertex algebras to complete the modular moonshine program of Borcherds and Ryba for proving an embedding of the sporadic group $F_3$ in $E_8(3)$.

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arXiv2012

Moonshine paths for 3A and 6A nodes of the extended E8-diagram

We continue the program to make a moonshine path between a node of the extended $E_8$-diagram and the Monster. Our theory is a concrete model expressing some of the mysterious connections identified by John McKay, George Glauberman and Simon Norton. In this article, we treat the 3A and 6A-nodes. We determine the orbits of triples $(x,y,z)$ in the Monster where $z\in 2B$, $x, y \in 2A \cap C(z)$ and $xy\in 3A \cup 6A$. Such $x, y$ correspond to a rootless $EE_8$-pair in the Leech lattice. For the 3A and 6A cases, we shall say something about the "half Weyl groups", which are proposed in the Glauberman-Norton theory. Most work in this article is with lattices, due to their connection with dihedral subgroups of the Monster. These lattices are $M+N$, where $M, N$ is the relevant pair of $EE_8$-sublattices, and their annihilators in the Leech lattice. The isometry groups of these four lattices are analyzed.

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arXiv2011

A new existence proof of the Monster by VOA theory

We use uniqueness of a VOA (vertex operator algebra) extension of $(V_{EE_8}^+)^3$ to a Moonshine type VOA to give a new existence proof of a finite simple group of Monster type. The proof is relatively direct. Our methods depend on VOA representation theory and are free of many special calculations which traditionally occur in theory of the Monster.

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arXiv2010

A moonshine path for 5A and associated lattices of ranks 8 and 16

We continue the program, begun in \cite{gl3cpath}, to make a moonshine path between a node of the extended $E_8$-diagram and the Monster simple group. Our goal is to provide a context for observations of McKay, Glauberman and Norton by realizing their theories in a more concrete form. In this article, we treat the 5A-node. Most work in this article is a study of certain lattices of ranks 8, 16 and 24 whose determinants are powers of 5.

math.GR↗

arXiv2009

A moonshine path from $E_8$ to the monster

One would like an explanation of the provocative McKay and Glauberman-Norton observations connecting the extended $E_8$-diagram with pairs of 2A involutions in the Monster sporadic simple group. We propose a down-to-earth model for the 3C-case which exhibits a logic to these connections.

math.GR↗

arXiv2008

$EE_8$-lattices and dihedral groups

We classify integral rootless lattices which are sums of pairs of $EE_8$-lattices (lattices isometric to $\sqrt 2$ times the $E_8$-lattice) and which define dihedral groups of orders less than or equal to 12. Most of these may be seen in the Leech lattice. Our classification may help understand Miyamoto involutions on lattice type vertex operator algebras and give a context for the dihedral groups which occur in the Glauberman-Norton moonshine theory.

math.RT↗

arXiv2005

On the uniqueness of the moonshine vertex operator algebra

It is proved that a vertex operator algebra is isomorphic to the moonshine VOA of Frenkel-Lepowsky-Meurman if it satisfies certain conditions. Our two main theorems establish a weak version of the FLM uniqueness conjecture for the moonshine vertex operator algebra. We believe that these are the first such results.

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arXiv2001

Virasoro frames and their Stabilizers for the E_8 lattice type Vertex Operator Algebra

The concept of a framed vertex operator algebra was studied in [DGH] (q-alg/9707008). This article is an analysis of all Virasoro frame stabilizers of the lattice VOA V for the E_8 root lattice, which is isomorphic to the E_8-level 1 affine Kac-Moody VOA V. We analyze the frame stabilizers, both as abstract groups and as subgroups of the Lie group Aut(V) = E_8(C). Each frame stabilizer is a finite group, contained in the normalizer of a 2B-pure elementary abelian 2-group in Aut(V). In particular, we prove that there are exactly five orbits for the action of Aut(V) on the set of Virasoro frames, settling an open question about V in Section 5 of [DGH]. The results about the group structure of the frame stabilizers can be stated purely in terms of modular braided tensor categories. Appendices present aspects of the theory of automorphism groups of VOAs. In particular, there is a result of general interest, on equivariant embeddings of lattices: embeddings of lattices into unimodular lattices which respect automorphism groups and definiteness. [DGH] C. Dong, R. Griess and G. Hoehn: "Framed vertex operator algebras, codes and the moonshine module", Comm. Math. Phys. 193 (1998), 407-448.

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arXiv1998

Rank one lattice type vertex operator algebras and their automorphism groups, II: E-series

Let L be the A_1 root lattice and G a finite subgroup of Aut(V_L), where $V_L$ is the associated lattice VOA (in this case, Aut(V) is isomorphic to PSL(2,\Bbb C)). The fixed point subVOA, V^G was studied in q-alg/9710017, which finds a set of generators and determines the automorphism group when G is cyclic (from the "A-series") or dihedral (from the "D-series"). In the present article, we obtain analogous results for the remaining possibilities for G, that it belong to the "E-series": G\cong Alt_4, Alt_5 or Sym_4.

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