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The McKay-Thompson series of Mathieu Moonshine modulo two

Thomas Creutzig, Gerald Höhn, Tsuyoshi Miezaki

TL;DR

This work analyzes the parity of coefficients in the McKay-Thompson series arising from Mathieu moonshine by embedding the problem in modular-form theory and employing Sturm-type verifications. It classifies which conjugacy classes yield odd coefficients (notably $7AB$, $14AB$, $15AB$, $21AB$, and $23AB$, with a refinement for $21AB$) and proves evenness for all other cases, including a careful handling of $11A$. These parity results yield precise multiplicity constraints for irreducible $M_{24}$-representations in the Mathieu moonshine module $K_n$, and together with representation-theoretic arguments confirm the Cheng–Duncan–Harvey conjecture in the Mathieu (UM) context. The methodology combines explicit eta-product modular forms, parity corrections via theta-difference functions, and Sturm’s theorem to obtain global parity statements and their representation-theoretic consequences.

Abstract

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.

The McKay-Thompson series of Mathieu Moonshine modulo two

TL;DR

This work analyzes the parity of coefficients in the McKay-Thompson series arising from Mathieu moonshine by embedding the problem in modular-form theory and employing Sturm-type verifications. It classifies which conjugacy classes yield odd coefficients (notably , , , , and , with a refinement for ) and proves evenness for all other cases, including a careful handling of . These parity results yield precise multiplicity constraints for irreducible -representations in the Mathieu moonshine module , and together with representation-theoretic arguments confirm the Cheng–Duncan–Harvey conjecture in the Mathieu (UM) context. The methodology combines explicit eta-product modular forms, parity corrections via theta-difference functions, and Sturm’s theorem to obtain global parity statements and their representation-theoretic consequences.

Abstract

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.

Paper Structure

This paper contains 7 sections, 3 theorems, 26 equations, 2 tables.

Table of Contents

  1. Introduction
  2. The non-even cases
  3. The even cases
  4. The cases $2A$, $3A$, $4B$, $5A$, $6A$, $8A$
  5. The case $11A$
  6. Multiplicities of Irreducibles
  7. McKay-Thompson series

Key Result

Theorem 1.1

The McKay-Thompson series as in GHV2EH determine a virtual graded $M_{24}$-module $K=\bigoplus_{n=-1}^{\infty} K_n\, q^{n/8}$. For $n\geq 0$, the $K_n$ are honest (and not only virtual) $M_{24}$-representations. Furthermore, $K_n$ can be decomposed as a direct sum of $M_{24}$-representations of the

Theorems & Definitions (8)

  • Theorem 1.1: Mathieu moonshine module
  • Theorem 1.2
  • Conjecture 1.1: CDH, Conj. 5.11
  • proof : Proof of Theorem \ref{['thm:main']} for the case $7AB$
  • proof
  • proof
  • Theorem 4.1
  • proof