Coefficients of McKay-Thompson series and distributions of the moonshine module
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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 monsterβs character table. We find analogous results for the other monster modules $V^{(-m)}$ and $W^\natural$ studied by Duncan, Griffin, and Ono.References
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Additional Information
- Hannah Larson
- Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
- MR Author ID: 1071917
- Email: hannahlarson@college.harvard.edu
- 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
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 4183-4197
- MSC (2010): Primary 11F03, 11F22, 20C34
- DOI: https://doi.org/10.1090/proc/13228
- MathSciNet review: 3531171