Monstrous Moonshine

In 1979, Conway and Norton discovered an unexpected intimate connection between the monster group M and the j-function. The Fourier expansion of j(tau) is given by


(OEIS A000521), where q^_=e^(2piitau) and tau is the half-period ratio, and the dimensions of the first few irreducible representations of M are 1, 196883, 21296876, 842609326, ... (OEIS A001379).

In November 1978, J. McKay noticed that the q^_-coefficient 196884 is exactly one more than the smallest dimension of nontrivial representations of the M (Conway and Norton 1979). In fact, it turns out that the Fourier coefficients of j(tau) can be expressed as linear combinations of these dimensions with small coefficients as follows:


Borcherds (1992) later proved this relationship, which became known as monstrous moonshine. Amazingly, there turn out to be yet more deep connections between the monster group and the j-function.

See also

j-Function, Monster Group

This entry contributed by David Terr

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Borcherds, R. E. "Monstrous Moonshine and Monstrous Lie Superalgebras." Invent. Math. 109, 405-444, 1992.Conway, J. H. and Norton, S. P. "Monstrous Moonshine." Bull. London Math. Soc. 11, 308-339, 1979.Sloane, N. J. A. Sequences A000521/M5477 and A001379 in "The On-Line Encyclopedia of Integer Sequences."

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Monstrous Moonshine

Cite this as:

Terr, David. "Monstrous Moonshine." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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