NOTES ON REPRESENTATIONS OF FINITE GROUPS Contents 1 ...

10 BEN MCKAYProof. For each g ∈ G, again pick a basis {v i } ⊂ V 1 of eigenvectors of ρ 1 (g),ρ 1 (g)v i = λ i v i ,and a basis {w j } ⊂ V 2 of eigenvectors of ρ 2 (g),ρ 2 (g)w j = µ j w j .Then using a G-invariant inner product on V 1 , we can define linear mapsφ ij ∈ Hom (V 1 , V 2 )byφ ij (v) = 〈v, v i 〉 w j .We then check thatρ(g)φ ij = µ j ¯λi φ ij .Exercise 13. Check the last step in the last proof.□8. Characters and class functionsDefinition 13. A class function for a finite group G is a function f : G → C so thatf(gh) = f(hg) for any g, h ∈ G.For example, the character of any representation is a class function.Exercise 14. Let C G be the vector space of all class functions on G, with usualaddition and scaling of functions. For G = Z n , what is the dimension of C G ?Definition 14. We define an inner product on class functions. If f 1 and f 2 are twoclass functions, define the inner product〈f 1 , f 2 〉 = 1 ∑f 1 (g)|G|¯f 2 (g) ,(where the bar is complex conjugate).g∈GExercise 15. Prove that the space C G of class functions is a finite dimensional innerproduct space, with this inner product.Theorem 4. If V 1 and V 2 are irreducible representations of a finite group G, saywith characters χ 1 and χ 2 , then{1 if V 1 is isomorphic to V 2 ,〈χ 1 , χ 2 〉 =0 otherwise.Proof. Let χ be the character of the representation Hom (V 1 , V 2 ). Recall χ = χ 2 ¯χ 1from lemma 10 on the preceding page.〈χ 1 , χ 2 〉 = 1 ∑χ 1 (g) ¯χ 2 (g)|G|so by lemma 7,= 1|G|g∈G∑χ (g)g∈G= dim Hom (V 1 , V 2 ) G ,