BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY

In about 1972 R. L. Griess and Fischer independently suggested the existence of a new sporadic simple group, which had a double cover of the “baby monster” discovered by Fischer as the centralizer of an involution. Calculations suggested that its smallest non-trivial complex representation probably had dimension 196883. “Moonshine” is the attempt to explain John McKay’s extraordinary observation that this number is almost equal to the coefficient 196884 of the elliptic modular function j(τ ) = q−1 + 744 + 196884q + 21493760q + · · · giving the j-invariant of the elliptic curve C/{1, τ} (where q = e ). At first, a common explanation was that there are many large numbers that turn up in mathematics, and a few of them will be almost equal just by coincidence. Shortly afterwards, John Thompson [T] pointed out that the next coefficient 21493760 of the elliptic modular function is equal to the sum of the dimensions of the first 3 irreducible representations of the monster, and similarly all the other coefficients seems to be simple linear combinations of dimensions of irreducible representations. He conjectured that this might be because the monster acts on a graded representation V whose piece of degree n is the coefficient of q in j(τ ) − 744, and suggested looking at the traces of other elements of the monster on this representation. (The coefficient 744 of q is a historical accident as adding an arbitrary constant to j still gives a function with similar properties. The most natural normalization is to set the constant term equal to 24, the number given by Rademacher’s infinite series for coefficients of the j function.) Conway and Norton [CN] carried out Thompson’s suggestion and found that the traces of all elements of the monster seem to be given by the coefficients of Hauptmoduls (roughly, functions invariant under a genus 0 congruence subgroup of SL2(R)). Atkin, Fong, and Smith [S] verified that the monster indeed has a representation with these properties by checking that the Hauptmoduls satisfied the necessary congruence and positivity conditions. The monster simple group was finally constructed by R. L. Griess [G] in 1982. Norton had observed a few years before that the monster probably had a commutative (but non-associative) algebra structure on its 196883 dimensional representation, and it was also known how to write the 196884 dimensional representation as a sum of three irreducible representations of a centralizer of an involution. The monster could therefore be constructed by adjusting the finite number of parameters that an algebra structure depends upon so that the algebra has an extra automorphism. While the idea is simple, the details are horrendous to carry out: John Conway told me that when he heard Griess had constructed the monster, he assumed that Griess must have found another way to do it, because he could not believe that anyone would be patient enough to finish this calculation.