9 Monstrous Moonshine
Monstrous moonshine connects the Monster group to modular functions via its dimension and the \(j\)-invariant.
9.1 Monster Dimension
The smallest faithful representation: \(196883\)
\(196883 = 47 \times 59 \times 71\)
\(196883 = L_8 \times (b_3- L_6) \times (b_3- 6)\)
\(47, 59, 71\) form an AP with common difference \(12 = \mathrm{dim}(G_2) - p_2\)
9.2 j-Invariant
\(j(\tau ) = q^{-1} + 744 + 196884q + \ldots \)
\(744 = N_{\mathrm{gen}} \times \mathrm{dim}(E_8) = 3 \times 248\)
\(744 = \mathrm{dim}(E_8) + \mathrm{dim}(E_8\times E_8) = 248 + 496\)