12 2. Basic notions of representation theory Let φ : V1 → V2 be a nonzero homomorphism of representations. Then: (i) If V1 is irreducible, φ is injective. (ii) If V2 is irreducible, φ is surjective. Thus, if both V1 and V2 are irreducible, φ is an isomorphism. Proof. (i) The kernel K of φ is a subrepresentation of V1. Since φ = 0, this subrepresentation cannot be V1. So by irreducibility of V1 we have K = 0. (ii) The image I of φ is a subrepresentation of V2. Since φ = 0, this subrepresentation cannot be 0. So by irreducibility of V2 we have I = V2. Corollary 2.3.10 (Schur’s lemma for algebraically closed fields). Let V be a finite dimensional irreducible representation of an algebra A over an algebraically closed field k, and let φ : V → V be an inter- twining operator. Then φ = λ · Id for some λ ∈ k (a scalar operator). Remark 2.3.11. Note that this corollary is false over the field of real numbers: it suﬃces to take A = C (regarded as an R-algebra) and V = A. Proof. Let λ be an eigenvalue of φ (a root of the characteristic poly- nomial of φ). It exists since k is an algebraically closed field. Then the operator φ−λ Id is an intertwining operator V → V , which is not an isomorphism (since its determinant is zero). Thus by Proposition 2.3.9 this operator is zero, hence the result. Corollary 2.3.12. Let A be a commutative algebra. Then every irreducible finite dimensional representation V of A is 1-dimensional. Remark 2.3.13. Note that a 1-dimensional representation of any algebra is automatically irreducible. Proof. Let V be irreducible. For any element a ∈ A, the operator ρ(a) : V → V is an intertwining operator. Indeed, ρ(a)ρ(b)v = ρ(ab)v = ρ(ba)v = ρ(b)ρ(a)v (the second equality is true since the algebra is commutative). Thus, by Schur’s lemma, ρ(a) is a scalar operator for any a ∈ A. Hence

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