Moonshine modules and a question of Griess

TL;DR

This paper explores the structure of modules under group actions linked to moonshine phenomena, revealing how moonshine induces partial orderings on irreducible representations and addressing a question by Griess.

Contribution

It provides a complete description of the asymptotic structure of graded modules with modular trace functions, connecting moonshine to representation orderings and verifying hypotheses for umbral moonshine.

Findings

Moonshine modules induce partial orderings on irreducible representations.

The hypothesis holds for umbral moonshine and certain vertex operator algebra automorphisms.

Asymptotic module structure is fully described under mild conditions.

Abstract

We consider the situation in which a finite group acts on an infinite-dimensional graded module in such a way that the graded trace functions are weakly holomorphic modular forms. Under a mild hypothesis we completely describe the asymptotic module structure of the homogeneous subspaces. As a consequence we find that moonshine for a group gives rise to partial orderings on its irreducible representations. This serves as a first answer to a question posed by Griess. In particular, we show that our hypothesis holds for umbral moonshine and for automorphism groups of certain vertex operator algebras.