Coefficients of McKay-Thompson series and distributions of the moonshine module (Q2816998)

This paper by develops asymptotic formulas for the coefficients of McKay-Thompson series defined for a ``tower of monster modules'' NEWLINE\[CARRIAGE_RETURNNEWLINEV^{(-m)} = \bigoplus_{n=-m}^\infty V_n^{(-m)} CARRIAGE_RETURNNEWLINE\]NEWLINE in order to answer questions surrounding the distribution of irreducible representations of the monster group \(\mathbb{M}\) in the graded pieces \(V_n^{(-m)}\). The author first obtains an ordering of the series based on the magnitude of their coefficients which is governed by a list of conjugacy classes for \(\mathbb{M}\). She then progresses to exploit this information to answer questions pertaining to the distribution of irreducible modules in the non-free part of \(V_n^{(-m)}\), when decomposed as a module over the group ring \(\mathbb{Z}[\mathbb{M}]\). In particular, it is found that the conjugacy class \(2A\) dictates the distributions of irreducible modules in the non-free part of \(V_n^{(-m)}\). This helps answer a question posed by Bob Griess and explained by \textit{J. F. R. Duncan} et al. [Res. Math. Sci. 2, Article ID 11, 57 p. (2015; Zbl 1380.11030)]. Additionally, a special case of the results developed in this paper occurs if \(m=1\), in which case \(V^{(-1)}\) is the well-known monster module vertex operator algebra \(V^\natural\).NEWLINENEWLINEThe introduction of the paper includes a brief description of Monstrous Moonshine, the pertinent definitions of the McKay-Thompson series for \(V^{(-m)}\), as well as three highlighted theorems that are proved later in the paper.NEWLINENEWLINEThe first theorem, Theorem \(1.1\), provides exact formulas for the coefficients in the aforementioned McKay-Thompson series. While similar to coefficients in Theorem \(8.12\) of the DGO paper mentioned above (and which is a good source for more information), the expressions given in Theorem \(1.1\) are useful to the author later in the paper. The descriptions of these coefficients are in terms of Atkin-Lehner involutions for a group, Kloosterman sums, and the Bessel function of the first kind. Using the asymptotics of the Bessel function, results developed pertaining to the Kloosterman sums (see Lemma \(3.1\)), and a series of lemmas obtained to determine which arithmetic progressions certain McKay-Thompson series are supported on (see Lemmas \(3.2\)--\(3.5\)), the author is then able to prove Theorem \(1.2\), which gives the asymptotic formulas for the McKay-Thompson coefficients.NEWLINENEWLINEFinally, the author turns her attention to analyzing multiplicities of irreducible representations in the non-free part of \(V_n^{(-m)}\). In this direction, asymptotic formulas are obtained for these multiplicities in Theorem \(1.3\). Additionally, Corollary \(1.4\) provides an interesting formula for the limit of the multiplicity of a representation in the non-free part divided by the sum of multiplicities of all other irreducible representations in the non-free part. The paper concludes with an appendix which includes necessary data pertaining to the conjugacy classes of \(\mathbb{M}\).