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

Larson, Hannah

doi:10.1090/proc/13228

Coefficients of McKay-Thompson series and distributions of the moonshine module
HTML articles powered by AMS MathViewer

by Hannah Larson PDF

Proc. Amer. Math. Soc. 144 (2016), 4183-4197 Request permission

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.

References

Lea Beneish and Hannah Larson, Traces of singular values of Hauptmoduln, Int. J. Number Theory 11 (2015), no. 3, 1027–1048. MR 3327854, DOI 10.1142/S1793042115500542
Richard E. Borcherds, Vertex algebras, Kac-Moody algebras, and the Monster, Proc. Nat. Acad. Sci. U.S.A. 83 (1986), no. 10, 3068–3071. MR 843307, DOI 10.1073/pnas.83.10.3068
Kathrin Bringmann and Ken Ono, Coefficients of harmonic Maass forms, Partitions, $q$-series, and modular forms, Dev. Math., vol. 23, Springer, New York, 2012, pp. 23–38. MR 3051180, DOI 10.1007/978-1-4614-0028-8_{3}
Jan H. Bruinier, Borcherds products on O(2, $l$) and Chern classes of Heegner divisors, Lecture Notes in Mathematics, vol. 1780, Springer-Verlag, Berlin, 2002. MR 1903920, DOI 10.1007/b83278
J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, $\Bbb {ATLAS}$ of finite groups, Oxford University Press, Eynsham, 1985. Maximal subgroups and ordinary characters for simple groups; With computational assistance from J. G. Thackray. MR 827219
J. H. Conway and S. P. Norton, Monstrous moonshine, Bull. London Math. Soc. 11 (1979), no. 3, 308–339. MR 554399, DOI 10.1112/blms/11.3.308
John F. R. Duncan and Igor B. Frenkel, Rademacher sums, moonshine and gravity, Commun. Number Theory Phys. 5 (2011), no. 4, 849–976. MR 2905139, DOI 10.4310/CNTP.2011.v5.n4.a4
John F. R. Duncan, Michael J. Griffin, and Ken Ono, Moonshine, Res. Math. Sci. 2 (2015), Art. 11, 57. MR 3375653, DOI 10.1186/s40687-015-0029-6
I. B. Frenkel, J. Lepowsky, and A. Meurman, A natural representation of the Fischer-Griess Monster with the modular function $J$ as character, Proc. Nat. Acad. Sci. U.S.A. 81 (1984), no. 10, , Phys. Sci., 3256–3260. MR 747596, DOI 10.1073/pnas.81.10.3256
I. Frenkel, J. Lepowsky, and A. Meurman, A moonshine module for the moster, Math. Sci. Res. Inst. Publ., vol. 3, Springer, New York, 1985, 231–273.
Koichiro Harada and Mong Lung Lang, The McKay-Thompson series associated with the irreducible characters of the Monster, Moonshine, the Monster, and related topics (South Hadley, MA, 1994) Contemp. Math., vol. 193, Amer. Math. Soc., Providence, RI, 1996, pp. 93–111. MR 1372718, DOI 10.1090/conm/193/02367
L. J. P. Kilford, Modular forms, Imperial College Press, London, 2008. A classical and computational introduction. MR 2441106, DOI 10.1142/p564
J. G. Thompson, Some numerology between the Fischer-Griess Monster and the elliptic modular function, Bull. London Math. Soc. 11 (1979), no. 3, 352–353. MR 554402, DOI 10.1112/blms/11.3.352