CHAPTER 1. GROUP RERRESENTATIONS 1.7. COMNUTANT AND ENDOMORPHISM ALGEBRAS 23 Proof, We prowe ouly the first assertion, leaving the scoond one for the reader. It is kaown from the thicory of vector spaces that kerθ is a subspace of V since θ is linear. So we oaly need to show closure under the action of G. But if for all g∈G. It follows that v∈ketθ, then for any g∈G i
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θ(g)=g(v)(θ is a G-lyomomorphism) where I is the approppiate Identity matrix and c∈C is any scalar, Now C is satisties the hypothesis of Corollary 1.6.6 (with X=Y ) and is not invertible the following result: and 50 gv ∈kerθ, as desired. Corollary 1.6.8 Let X be an imcducible matrix representation of G oner the Theorem 1.6.5 (Schur's Lemma) Let V and W be two irreducible G modules. If θ:V→W is a G-homomiorphasm, then either 1. θ is a G-isomorahism, of 14. Prove the following converse of Schur's lemma. Let X be a represen- 2. θ is the zero map. tation of G over C with the property that only scalar multiples cI commute with X(g) for all g∈G. Prove that X is irreducible. Proof. Since V is irreducible and ker θ is a submodule (by the previous 15. Let X and Y be representations of G. The inner tensor product, X ⊗
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Y, proposition), we must have elther ker θ={0} or ker θ=V. Similarly, the assigns to each g∈G the matrix irreciucibility of W implies thnt im θ={0} or W. If kerθ=V or imθ={0}, (X ⊗
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Y)(g)=X(g)⊗Y(g). then θ must be the zero map. On Is is interesting to note that Schur's lenma continues to be valid over (a) Verify that X ⊗
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Y is a representation of G. arbitrary fields and for infinite groups. In fict, the proof we just gave still (b) Show that if X,Y, and X ⊗
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Y have characters denoted by χ,ψ, works: The matrix yersion is also true in this more general setting. and χ ⊗
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ψ, respectively, then (χ ⊗
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ψ)(g)=χ(g)ψ(g). Corollary 1.6.6 Let X and Y be tuo irreducible matrix representations of (c) Find a group with irreducible representations X and Y such that G. If T is any matrix such that TX(g)=Y(g)T for all g∈G, then either X ⊗
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Y is not irreducible. 1. T is moertible, or 2. T is the zero matrix. We also have an analogue of Schur's leama in the case where the range module is not irreducible. This result is conveniently expressed in terms of the vector spece Hom (V;W) of all G-homomorphisms from V to W. Corollary 1.6.7 Let V and W be two G-modules with V being irredacible. Then dim Hom (V,W)=0 if and only if W contains no submodule isomorphic to K −
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When the field is C, however, we can say more. Suppose that T is a matrix suel that TX(g)=X(g)T