3.1 Representation and Character Theory We can consider any finite group as a finite subgroup of the group of n×ninvertible matrices acting on V, where V is a finite-dimensional complex vector space. If we consider Gin this way, we are looking at a (complex linear finite-dimensional) representation of Gwhich is a pair (ρ, V), where ρ : G→GL(V) is a homomorphism (a function that preserves the group operation) [11]. Officially, we can define a representationof Gover F (where we will think of F as C={a+bi : a, b ∈R, i2 =−1}, the complex numbers) as a homomorphismρ fromG to GL(n, F) for some n[9]. Note that GL(n, F) is the general linear group of n×ninvertible matrices with entries fromF. We define the dimension (or degree) of the representation to be the dimension of V [11]. Let us think about the square group we looked at earlier. Here is an example of a homomorphism, ρ, from the square group to GL(2, C): g ρ(g) g ρ(g) id 1 0 0 1! x 1 0 0 −1! y 0 1 −1 0! yx 0 −1 −1 0 ! y2 −1 0 0 −1! y2x −1 0 0 1! y3 0 −1 1 0 ! y3x 0 1 1 0! . Note that if we multiply these matrices, they behave in the same way as their corresponding elements in D4. For example, if we multiple the matrices corresponding to y and x we get 0 1 −1 0! 1 0 0 −1! = 0 −1 −1 0 !, which is the matrix corresponding to yx. Groups can be partitioned into conjugacy classes. Given an element a ∈G, we define the conjugacy class of a to be cl(a) ={b ∈G|b =c−1ac, c ∈G}. Here is another example of a matrix representation of the square group that is conjugate to Page 30 Riley • An Overview of Monstrous Moonshine
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