5 Monstrous Moonshine Based on the connection between the coefficients of the J-function and the dimensions of the irreducible representations of the Monster, John Thompson made the following conjecture: Conjecture 1: “There is a (somehow) natural infinite-dimensional graded representation of the monster group (ρ\, V \ =⊕i≥−1V \ i ), such that each graded part V \ i is finite dimensional, and such that: ˜ J(τ) = X i≥−1 dim(V\ i )q i. Equivalently, the elements of Mact naturally as (infinite) matrices on an infinite dimensional graded vector space (these matrices are block-diagonal with every block of finite size), and the graded-dimension of V\ is the q-expansion of the normalized J-function” [11]. Note that in this definition, ˜ J(τ) is the same as what we have been referring to as J(τ), namely, the normalized J-function. This conjecture is quite intense, so some explanation is in order. Similar to the representations we were considering earlier, this conjecture says that there is an infinitedimensional representation of the Monster where ρ\ is our homomorphism and V \ is our vector space. The vector space we are considering is graded, which means that the vector space is a direct sum of subspaces which are indexed by some set. A direct sum is similar to a direct product except that all but finitely many terms must be 0. Direct sums are used in commutative groups where we can consider our binary operation using additive notation [4]. The notation for a direct sum is ⊕. In this conjecture, each of the subspaces has finite dimension. Thompson’s conjecture says that there is an infinite dimensional graded module for the Monster whose graded submodule corresponding to i has dimension the cofficient of qi in the J-function. The conjecture can also be thought of in terms of matrices. If we consider the elements of Mas matrices that act on an infinite dimensional graded vector space, the graded dimension is the q-expansion of J(τ). If we look at representations of groups, we are simply looking at the trace of ρ(e), the identity element. For this reason, Thompson suggested that the series T[g] = X i≥−1 Tr(ρM(g)V\ i )qi = 1 q + ∞ X n=0 Hn([g])q n should be studied. These series (which are now called McKay-Thompson series) exist for each conjugacy class. So there is a separate formula for each conjugacy class in M. Page 46 Riley • An Overview of Monstrous Moonshine
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