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, with no obvious reason to touch the j-function, yet the first coefficients of j split into sums of dimensions of its representations. The study of that connection is called monstrous moonshine. The figure below decomposes the first three.
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 them. 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 is cyclic of prime order, alternating (An, n ≥ 5), of Lie type (16 infinite families over finite fields), or one of 26 sporadic groups.
The sporadic groups were found between 1861 and 1976, from M11 (order 7920) to the Monster, much the largest. Twenty of the 26 occur inside the Monster as quotients of its subgroups; the other six are called pariahs.
Fischer and Griess predicted the Monster independently in 1973, as a simple group in which some involution has centraliser a double cover of Fischer's Baby Monster. Griess computed from Thompson's order formula that such a group would have order
Norton showed that it 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, the Griess algebra, which as a Monster module is 1 ⊕ 196883. McKay had noticed in 1978 that this total is c(1).
The order has exactly 15 prime factors: 2 divides it 46 times, and the largest primes, 59 and 71, once each. In a 1974–75 seminar paper Andrew Ogg showed that the primes p for which X₀(p)+ has genus zero are exactly those for which every supersingular j-invariant in characteristic p, a j whose curve has no points of order p even over the algebraic closure, lies in 𝔽p rather than only in 𝔽p². In a lecture on 14 January 1975 Jacques Tits had described the conjectured Monster and its order, and Ogg saw that its primes are exactly these. He offered a bottle of Jack Daniel's to anyone who could explain the coincidence. Borcherds' 1992 proof of moonshine gives a route from the Monster to the genus-zero property but not back, and Duncan and Ono (2014) still describe the coincidence as unexplained.
For each prime up to 149, the supersingular j-invariants in characteristic p, computed as roots of the Hasse polynomial: solid if in 𝔽p, outlined if only in 𝔽p². Shaded columns have every one in 𝔽p. They are exactly the 15 primes dividing |𝕄|, whose exponents stand above.
A finite group's character table has one row per irreducible representation (irrep) and one column per conjugacy class, each entry a trace, and a group with k classes has exactly k irreps. The Monster has 194. Fischer, Livingstone and Thorne computed its table in Birmingham in 1978, assuming an irreducible character of degree 196883, before the construction showed that the group exists; it was printed in the ATLAS of Finite Groups (1985).
The dimension of an irrep ρk is χk(1), and for any finite group the squares of the dimensions add up to the order:
So no dimension exceeds √|𝕄| ≈ 9.0 × 1026. The largest is
whose square alone is about 8% of the group order.
On one side is the q-expansion of j,
and on the other the first irrep dimensions of 𝕄,
In 1978 McKay noticed that c(1) = 1 + 196883 = dim ρ1 + dim ρ2 and told John Thompson. The next coefficients split the same way, with multiplicities 0, 1 or 2:
A sum of dimensions on its own proves little. Since dim ρ1 = 1, every positive integer is a non-negative combination of irrep dimensions, in many ways: 417 444 627 ways for c(4). Even with small multiplicities, c(4) = 20 245 856 256 is both 3·1 + 3·196 883 + 21 296 876 + 2·842 609 326 + 18 538 750 076 and 2·1 + 3·196 883 + 2·21 296 876 + 842 609 326 + 19 360 062 527.
Thompson saw a sharper test. If the coefficients are dimensions of representations Vn, every element g of 𝕄 has a trace on each of them, and the traces should also assemble into functions worth studying. Conway and Norton found them: for g in the class 2A the series is t + 24 + 4096/t with t = (η(τ)/η(2τ))24, which is unchanged by Γ0(2) and by τ ↦ −1/(2τ). The trace of g on a sum of irreps is the same sum of character values, so each class adds one equation.
Ways to write c(n) as a non-negative sum of Monster irrep dimensions, counted exactly: by dimension alone, then also matching the trace on one more class to its McKay–Thompson series (from an eta quotient, with ATLAS character values), then all six. Select a row to check it class by class.
One class, 2A, already leaves a single decomposition for c(1) to c(4), and for c(4) it is the second of the two above; for c(5) it leaves two. The six classes together leave exactly one each time, and it has the small multiplicities. Thompson's reading was that there is a graded Monster module
with V0♮ = 0, 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 pieces. Part 13 states the Conway–Norton conjecture, that the graded trace of every element is a Hauptmodul of a genus-zero group, and builds V♮ from the Leech lattice. The lattices themselves are covered in the Exceptional Atlas series.