Coefficients of McKay-Thompson series and distributions of the moonshine module

TL;DR

This paper investigates the asymptotic distribution of irreducible representations in moonshine modules by analyzing McKay-Thompson series coefficients, revealing the dominant role of the conjugacy class 2A in the monster group.

Contribution

It provides asymptotic formulas characterizing the distribution of irreducible components in moonshine modules, especially focusing on the non-free part influenced by class 2A.

Findings

Asymptotic distribution of non-free parts is dominated by conjugacy class 2A.

Orderings of McKay-Thompson series by coefficient magnitude reflect contributions to representation distribution.

Results extend to other monster modules like V^{(-m)} and W^ atural.

Abstract

In a recent paper, Duncan, Griffin and Ono provide exact formulas for the coefficients of McKay-Thompson series and use them to find asymptotic expressions for the distribution of irreducible representations in the moonshine module $V^\natural = \bigoplus_n V_n^\natural$. Their results show that as $n$ tends to infinity, $V_n^\natural$ is dominated by direct sums of copies of the regular representation. That is, if we view $V_n^\natural$ as a module over the group ring $\mathbb{Z}[\mathbb{M}]$, the free-part dominates. A natural problem, posed at the end of the aforementioned paper, is to characterize the distribution of irreducible representations in the non-free part. Here, we study asymptotic formulas for the coefficients of McKay-Thompson series to answer this question. We arrive at an ordering of the series by the magnitude of their coefficients, which corresponds to various contributions to the distribution. In particular, we show how the asymptotic distribution of the non-free part is dictated by the column for conjugacy class 2A in the monster's character table. We find analogous results for the other monster modules $V^{(-m)}$ and $W^\natural$ studied by Duncan, Griffin, and Ono.