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

TL;DR

This paper explores the connections between certain nodes of the extended E8 diagram and the Monster group, using lattice theory and group actions to deepen understanding of moonshine phenomena.

Contribution

It provides a concrete model linking E8 nodes to the Monster, analyzing lattice structures and group actions for the 3A and 6A cases in moonshine theory.

Findings

Determined orbits of triples in the Monster related to 3A and 6A nodes.

Analyzed isometry groups of specific lattices connected to the Monster.

Explored the role of half Weyl groups in the Glauberman-Norton theory.

Abstract

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.