NOTES ON REPRESENTATIONS OF FINITE GROUPS Contents 1 ...

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6 BEN MCKAYso that each V i is not isomorphic to any V j for i ≠ j. The summands n i V i areuniquely determined, as are the multiplicities n i , up to ordering. The various V iare uniquely determined up to isomorphism.Proof. We know that we can split V into irreducible representations by theorem 1on the preceding page. Suppose that we have two splittings as above, sayV = n 1 V 1 ⊕ n 2 V 2 ⊕ · · · ⊕ n k V k , = p 1 W 1 ⊕ p 2 W 2 ⊕ · · · ⊕ p l W l .The identity map V → V yields a map on each factor V i , and either identifies itwith a factor W j , or maps it to that factor by the 0 linear map.□Example 3. Careful: take any group G and the trivial G representation on V = C 2 .Then we can split C 2 = C ⊕ C. But we could also (for example) let V 1 be the spanof the vector ( 11)and let V 2 be the span of the vector( ) 1.−1Then V = V 1 ⊕ V 2 . So V has many decompositions into G-invariant subspaces, but(by the theorem) only one into maximal sums of irreducible subspaces.Definition 8. If ρ 1 : G → GL (V 1 ) and ρ 2 : G → GL (V 2 ) are two representations ofa finite group, then W = Hom (V 1 , V 2 ) (the set of all linear maps V 1 → V 2 ) hasG-representationρ: G → GL (W ) ,given by taking ρ(g)φ to be the linear map(ρ(g)φ) (v) = ρ 2 (g)φ ( ρ 1 (g) −1 v ) .Exercise 5. Prove that this makes Hom (V 1 , V 2 ) into a G-representation.Definition 9. A morphism of representations is a linear map φ: V 1 → V 2 so thatφ (ρ 1 (g)v) = ρ 2 φ(v),for all vinV 1 and g ∈ G. Let Hom G (V 1 , V 2 ) be the set of all morphisms of representationsV 1 → V 2 .Exercise 6. Prove that Hom G (V 1 , V 2 ) is precisely the set of vectors in Hom (V 1 , V 2 )which are invariant under all elements of G, i.e.Hom G (V 1 , V 2 ) = Hom (V 1 , V 2 ) G .Lemma 6. If V and W are two representations of a finite group G, say with eachsplitting into irreducibles asandV = p 1 V 1 ⊕ p 2 V 2 ⊕ . . . p k V k ,W = q 1 W 1 ⊕ q 2 W 2 ⊕ . . . q l W l ,then Hom G (V, W ) splits into irreducibles asHom G (V, W ) = ⊕ ijp i q j Hom (V i , W j ) ,
NOTES ON REPRESENTATIONS OF FINITE GROUPS 7where the direct sum is a sum over pairs i and j for which V i and W j are isomorphic.Moreover, Hom (V i , W j ) is then 1-dimensional.Proof. From theorem 3 on page 5, we have unique splittings into irreducibles. Fromtheorem 2 on page 5, any linear map φ ∈ Hom (V, W ) must split into a sum ofisomorphisms and 0 maps. Moreover, each isomorphism is unique up to scaling, a1-dimensional representation Hom (V i , W j ).□7. CharactersDefinition 10. The trace of a square (say n × n) matrix A isthe sum of the diagonal entries.tr A = A 11 + A 22 + · · · + A nn ,Exercise 7. Prove that tr(AB) = tr(BA) for any n × n matrices A and B.Definition 11. The traceof a linear map φ: V → V on a finite dimensional vector space V is defined bytr φtr φ = tr Awhere A is the matrix of φ in some basis.Exercise 8. Prove that tr φ does not depend on the choice of basis.Choosing a basis in which A is in Jordan normal form, we see that tr φ is thesum of the eigenvalues, counted with multiplicities.Definition 12. If ρ: G → GL (V ) is a representation, its character is the functionχ (also written as χ ρ ) given byχ(g) = tr ρ(g).Exercise 9. Let G be the group of permutations of 1, 2, 3. Write out each permutation,and find the value of the character of the permutation representation on eachpermutation.Exercise 10. Prove that if G is the group of permutations of 1, 2, . . . , n then thecharacter of the permutation representation is given by χ(g) equal to the numberof numbers from among 1, 2, 3, . . . , n which are fixed by g.Exercise 11. If χ is the character of an n-dimensional representation of a finitegroup G, then χ(1) = n.Lemma 7. The average of the character of a representation is the dimension ofthe set of fixed vectors. In other words, if ρ: G → GLV is a representation withcharacter χ, then1 ∑χ(g) = dim V G .|G|g∈GProof. Let p: V → V be the averaging operator. By lemma 5 on page 5, dim V G =tr p. Butp(v) = 1 ∑ρ(g)v,|G|g∈G
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