g ρ(g) σ(g) Tr(ρ(g))) = Tr(σ(g))) =χ(g) id 1 0 0 1! 1 0 0 1! 2 y 0 1 −1 0! i 0 0 −i! 0 y2 −1 0 0 −1! −1 0 0 −1! −2 y3 0 −1 1 0 ! −i 0 0 i! 0 x 1 0 0 −1! 0 1 1 0! 0 yx 0 −1 −1 0 ! 0 i −i 0! 0 y2x −1 0 0 1! 0 −1 −1 0 ! 0 y3x 0 1 1 0! 0 −i i 0 ! 0 Comparison of ρ and σ Note that the characters are the same for both representations, as expected. Basically, when we are looking at representations of a group, we can think of it as looking at a homomorphism from a group into GL(n, C). Alternatively, we can think of representations as FG-modules. Note that F is a field, Gis a group, and an FG-module is similar to a vector space. Since these ways of thinking of representations are equivalent, we will present a theorem that refers to modules and then apply it to homomorphisms. Maschke’s Theorem: “Let G be a finite group, let F be R or C, and let V be an FGmodule. If U is an FG-submodule of V, then there is an FG-submodule W of V such that V =U⊕W” [9]. Alternatively, “Let G be a finite group and F a field whose characteristic does not divide |G|. Then every F[G]-module is completely reducible” [8]. This theorem tells us that we can break modules down into a direct sum of submodules. In other words, if we take any one of these (complex linear finite-dimensional) modules, it will be isomorphic to a direct sum Page 32 Riley • An Overview of Monstrous Moonshine
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