On Majorana representations of the group $3^2{:}2$ of 3C-pure type and the corresponding vertex operator algebras

TL;DR

This paper investigates specific vertex operator algebras generated by Ising vectors with a group symmetry of shape 3^2{:}2, providing a classification, structural analysis, and explicit construction within a lattice VOA.

Contribution

It classifies Griess algebras generated by Ising vectors with a 3^2{:}2 symmetry and constructs explicit examples inside the lattice VOA V_{E_8^3}.

Findings

Griess algebra is uniquely determined up to isomorphism.

Constructs explicit VOA example inside V_{E_8^3}.

Analyzes the structure of the associated vertex operator algebra.

Abstract

In this article, we study Griess algebras and vertex operator subalgebras generated by Ising vectors in a moonshine type VOA such that the subgroup generated by the corresponding Miyamoto involutions has the shape $3^2{:}2$ and any two Ising vectors generate a 3C subVOA $U_{3C}$. We show that such a Griess algebra is uniquely determined, up to isomorphisms. The structure of the corresponding vertex operator algebra is also discussed. In addition, we give a construction of such a VOA inside the lattice VOA $V_{E_8^3}$, which gives an explicit example for Majorana representations of the group $3^2{:}2$ of 3C-pure type.