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


Author: Hannah Larson
Journal: Proc. Amer. Math. Soc. 144 (2016), 4183-4197
MSC (2010): Primary 11F03, 11F22, 20C34
DOI: https://doi.org/10.1090/proc/13228
Published electronically: June 3, 2016
MathSciNet review: 3531171
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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.


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Additional Information

Hannah Larson
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
Email: hannahlarson@college.harvard.edu

DOI: https://doi.org/10.1090/proc/13228
Received by editor(s): August 15, 2015
Received by editor(s) in revised form: January 3, 2016
Published electronically: June 3, 2016
Communicated by: Kathrin Bringmann
Article copyright: © Copyright 2016 American Mathematical Society