Using the character table of the Monster, in 1979 Conway and Norton added to Thompson’s conjecture. This new conjecture, Conway-Norton’s Monstrous Moonshine, can be stated as follows: Conjecture: Conway-Norton’s Monstrous Moonshine: “There is a (somehow) natural infinite-dimensional graded representation (ρM, V \ =⊕V\ i ) of the monster group, with finite dimensional graded parts V\ i , and such that for each conjugacy class [g] of the monster, the series T[g] is the q-expansion of the normalised Hauptmodul of some subgroup Γ[g] of PSL(2, R) commensurable with PSL(2, Z)” [11]. This definition leads back to our previous discussion of modular functions. Conway and Norton also computed Tr(ρM(g)V\ i ) for all g ∈Mand i ≤10, as well as verified that the functions fromGto C defined by g 7→Tr(ρM(g)V\ i ) are characters of some representations of M[11]. Using an IBM 370/158 computer, Atkin, Fong, and Smith showed in 1979 that the Hn class function are truly characters of M. They did this by reducing the infinite number of verifications down to a finite one by using results like Brauer’s characterization of characters [11]. Frenkel, Lepowsky, and Meurman worked to nail down precisely what was happening between the J-function and the Monster. They built a vertex operator algebra that led to the Monster by modifying the definition of vertex algebra given by Borcherds. This ended up being related to constructions in string theory. As Frenkel, Lepowsky, and Meurman say in their book, Vertex Operator Algebras and the Monster [6] (which was a follow-up to their ground-breaking article, “A Natural Representation of the Fischer-Griess Monster with the Modular Function J as Character” [5]), Combining this result [that V\ has the structure of a vertex operator algebra] with the rest of our earlier work, including triality, we are able to prove the following: (A) The Z-graded space V\ carries an explicitly defined vertex operator algebra structure with graded dimension J(τ) and rank 24, and which acts irreducibly on itself. (B) The Monster acts faithfully and homogeneously on the Z-graded space V\, preserving the vertex operator algebra structure as in (25) (with v ∈ V\) and fixing the elements 1 and ω [6]. Channels • 2022 • Volume 6 • Number 2 Page 47
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