By Knuppel F.

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**Example text**

Here are two examples: Example 1 (Z4 over the reals). Consider the abelian group Z 4 and irreps as real representations. Our analysis below will show there are three. Let [1] be the generator of Z 4 . The irreps are then characterized by U([1]) since [1] generates Z 4 . Then U1([1]) = 1, U2([1]) = -1, U3([1]) = ( -~ ~). The interesting aspect here is that U3 is irreducible, even though all irreps over the complexes are one-dimensional because Z 4 is abelian. , acting on C 2 ), U3 has the invariant subspaces generated by and CJ G) But neither lies in JR 2 .

21 II. 22 FUNDAMENTALS OF GROUP REPRESENTATIONS Proof. Let ( , ) 0 be an inner product on V. Define (v, w) 1 = o(G) L (U(g)v, U(g)w) 0. gEG It is easy to see that ( ·, ·) is also an inner product. Moreover, 1 (U(h)v, U(h)w) = o(G) L (U(g)U(h)v, U(g)U(h)w)o gEG 1 = o(G) L(U(gh)v, U(gh)w)o gEG 1 = o(G) L(U(g)v,U(g)w)o gEG = (v, w), where we used the fact that for h fixed, g ~ gh is a bijection, so as g runs through G, so does gh. D Example. Not only does the proof depend on G finite, so does the result.

2 Characters, class functions, and conjugacy classes One (but not the only) goal of this section is to show that #(G)= #(conjugacy classes of G). Definition. 1) --4 C so that all x, y E G. 2) We let Z(G) be the set of class functions on G. Thus, class functions are constant on conjugacy classes, but the values on distinct classes are independent. 3) dim Z(G) =#(classes of G). 1. Z(G) is the center of A( G); that is, g * f for all g E A( G). f E Z(G) if and only if f *g = Proof. 2) is equivalent to f(xz) = f(zx) all x, z E G.