The McKay-Thompson series of Mathieu Moonshine modulo two

TL;DR

This paper investigates the parity of coefficients in the McKay-Thompson series related to Mathieu moonshine and proves a conjecture connecting to Umbral moonshine, enhancing understanding of moonshine phenomena.

Contribution

It provides a detailed analysis of the parity of coefficients and confirms a conjecture linking Mathieu moonshine to Umbral moonshine.

Findings

Parity of McKay-Thompson series coefficients determined

Conjecture of Cheng, Duncan, and Harvey proved for Mathieu moonshine

Deeper insight into the structure of Mathieu moonshine coefficients

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.