Part 12 of 14. The Monster is the largest sporadic simple group. The coefficients of j split into sums of its representation dimensions, which is where moonshine begins.
Part 11 ended with McKay's observation that 196 884 = 1 + 196 883. The Monster comes from finite group theory, and there is no obvious reason for it to be related to the j-function. Yet the first Fourier coefficients of j split into sums of dimensions of its representations, and the study of that connection is called monstrous moonshine. This part introduces the group, its order and its character table, and then asks how much the coincidence shows. The figure below decomposes the first three coefficients.
Each bar is one term, dim ρk times its multiplicity, and the stacked bars total the exact j-coefficient. From c(1) to c(3) more irreps enter and the multiplicities stay small.
A finite group is simple if its only normal subgroups are the identity and the whole group. Simple groups play the role of primes: by the Jordan–Hölder theorem every finite group is built from simple groups by iterated extensions, so classifying them is the central problem of finite group theory.
The Classification of Finite Simple Groups, completed over roughly 1955–2004 in tens of thousands of pages by about a hundred authors, says that every finite simple group belongs to one of four families:
The sporadic groups were found between 1861 and 1976. They range from M11 (order 7920) to the Monster. The Monster is the largest by a wide margin; 20 of the 26 sporadics occur inside it as subquotients, that is, as quotients of its subgroups (the "happy family"). The other six are the pariahs.
The Monster was predicted independently by Fischer and Griess in 1973, as a simple group in which some involution has centraliser a double cover of Fischer's Baby Monster. Within months Griess had computed, from Thompson's order formula, that such a group would have order
Norton later showed that such a group would have a faithful representation of dimension 196883, the smallest possible. Griess announced a construction by hand in January 1980 and published it in 1982: the Monster is the automorphism group of a 196884-dimensional commutative, non-associative real algebra, now called the Griess algebra. As a Monster module the Griess algebra is 1 ⊕ 196883, scalars plus the smallest faithful representation, and McKay had already noticed in 1978 that this total is c(1).
The Monster's order has exactly 15 distinct prime factors. The largest power is 246, and the largest primes, 59 and 71, each divide it once.
Prime factorisation of |𝕄|: one bar per prime p, with height equal to its exponent. Every bar is orange because every one of these primes is a supersingular prime, one of the 15 primes p for which all supersingular j-invariants in characteristic p lie in 𝔽p, not just 𝔽p². That these are exactly the primes dividing |𝕄| is Ogg's observation of 1975, described below.
In a 1974–75 seminar paper, Andrew Ogg showed that the primes p for which every supersingular j-invariant lies in 𝔽p are exactly the primes for which the curve X₀(p)+ has genus zero:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71
These are the supersingular primes. In a lecture on 14 January 1975, Jacques Tits had described the conjectured Monster and its order, and Ogg noticed that its prime divisors are exactly these fifteen. In a remark at the end of the paper he offered a bottle of Jack Daniel's to anyone who could explain the coincidence. The match links the Monster's internal structure to the arithmetic of elliptic curves, which is where j lives. Borcherds' 1992 proof of moonshine gives a route from the Monster to the genus-zero property but not back again, and Duncan and Ono (2014) still describe the coincidence as unexplained; the bottle is unclaimed.
With about 8 × 10⁵³ elements, the Monster cannot be stored element by element. Computations with it go through its representations, its character table and its subgroups.
A finite group's character table has one row per irreducible representation (irrep) and one column per conjugacy class, and each entry is a trace. A group with k conjugacy classes has exactly k complex irreps, so the table is a k × k matrix of algebraic integers whose rows and columns both satisfy orthogonality relations.
The Monster has 194 conjugacy classes and so 194 irreps. Its character table was computed by Fischer, Livingstone and Thorne in Birmingham in 1978, on the assumption that the group existed with an irreducible character of degree 196883, and was later printed in the ATLAS of Finite Groups (1985). The table came before Griess's 1980 construction showed that the group exists.
Each irrep ρk is a complex vector space on which 𝕄 acts by matrices. Its dimension is χk(1), the trace of the identity, and for any finite group the squares of the dimensions add up to the group order:
For the Monster, the dimensions of the 194 irreps range from 1 to 258 823 477 531 055 064 045 234 375 ≈ 2.6 × 1026. The first five are the ones that decompose the j-function's Fourier coefficients:
| ρk | dim ρk | log₁₀ |
|---|
The first five Monster irrep dimensions. ρ₁ is the trivial representation and ρ₂ the smallest faithful one. After ρ₂ each dimension is 20 to 110 times the one before.
The sum-of-squares identity caps every dimension: no irrep has dim ρk larger than √|𝕄| ≈ 9.0 × 1026. The largest Monster irrep, ρ194, has dimension
so its square alone accounts for about 8% of the group order. The other 193 squares make up the rest.
The chart puts the first five irrep dimensions on a log scale beside the j-coefficients they match. At this scale dim ρ₂ = 196883 and c(1) = 196884 look identical.
Monster irrep dimensions (purple) and j-coefficients (blue) on a log scale. The difference of one between dim ρ₂ and c(1) is the trivial representation ρ₁.
On one side is the q-expansion of j,
and on the other the first five irrep dimensions of 𝕄,
In 1978 McKay noticed that c(1) = 196884 = 1 + 196883 = dim ρ1 + dim ρ2, and told John Thompson. The next coefficients split the same way:
In these three, each coefficient is a sum of irrep dimensions with multiplicities 0, 1 or 2, and the higher irreps enter one at a time.
A sum of dimensions on its own proves little. Since dim ρ1 = 1, every positive integer is some non-negative combination of irrep dimensions, in many ways once ρ1 is allowed to pad the sum. Even the decompositions with small multiplicities stop being unique at c(4): 20 245 856 256 = 3·1 + 3·196 883 + 21 296 876 + 2·842 609 326 + 18 538 750 076, and also = 2·1 + 3·196 883 + 2·21 296 876 + 842 609 326 + 19 360 062 527. What made the pattern hard to dismiss was that the natural small sums kept appearing, and that Thompson saw a sharper test. If these numbers are dimensions of actual representations, then every element g of 𝕄 has a trace on each of them, and those traces should also assemble into functions worth studying.
Write J(τ) = j(τ) − 744 = q−1 + 0 + 196884q + … for j without its constant term. When coefficients become dimensions or traces, read them as coefficients of J unless the text says otherwise.
Thompson's reading was that these decompositions are what you would see if there were an infinite-dimensional graded vector space
where each Vn♮ is a finite-dimensional representation of 𝕄 and dim Vn♮ = c(n) for n ≠ 0 and dim V0♮ = 0, the coefficients of J. (Here Vn♮ is indexed by the power of q; Part 13 explains how this differs by one from the usual weight.) Splitting c(n) into irrep dimensions is then splitting Vn♮ into irreducible summands, and J = j − 744 is the graded dimension of a Monster module.
Part 13 makes this precise. It states the Conway–Norton conjecture, which asks that the graded traces of every Monster element be Hauptmoduls of genus-zero groups as well as that the dimensions match, and then builds V♮ from the Leech lattice. The lattices themselves (E8 and Leech) are covered in the Exceptional Atlas series.