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Schur's lemma states that any intertwining operator between irreducible representations is either zero or an isomorphism. Over an algebraically closed field, any self-intertwining operator of an irreducible representation is a scalar multiple of the identity. This seemingly simple result has enormous consequences: it constrains the structure of Hom_G spaces, underpins the orthogonality relations for characters, and is the single most-used tool in representation theory.
Schur's lemma is the workhorse of representation theory. It comes in two parts. Part 1: If ρ: G → GL(V) and σ: G → GL(W) are irreducible representations and T: V → W is a G-equivariant linear map (meaning Tρ(g) = σ(g)T for all g), then T is either the zero map or an isomorphism. Part 2 (over an algebraically closed field like ℂ): If T: V → V is a G-equivariant endomorphism of an irreducible representation, then T = λI for some scalar λ.
The proof of Part 1 is elegant and short. The key observation is that ker(T) ⊆ V and im(T) ⊆ W are both G-invariant subspaces. For the kernel: if v ∈ ker(T), then T(ρ(g)v) = σ(g)(Tv) = σ(g)(0) = 0, so ρ(g)v ∈ ker(T). Similarly, the image is invariant. Since V is irreducible, ker(T) is either {0} or V; since W is irreducible, im(T) is either {0} or W. If T ≠ 0, the kernel must be {0} (T is injective) and the image must be W (T is surjective), so T is an isomorphism. There is no room for anything in between.
Part 2 uses algebraic closure. Over ℂ, the operator T: V → V has at least one eigenvalue λ. The map T − λI is still G-equivariant (since λI commutes with everything), and it has a nontrivial kernel (the λ-eigenspace). By Part 1, T − λI must be zero, so T = λI. Over ℝ, this argument fails because real operators need not have real eigenvalues — for instance, a 90° rotation has eigenvalues ±i.
The consequences are far-reaching. For abelian groups over ℂ, every ρ(g) commutes with the entire representation and is therefore scalar by Part 2. This forces all irreducible representations to be one-dimensional — a complete classification in one stroke. For non-abelian groups, Schur's lemma constrains the algebra of intertwining operators (the endomorphism ring of an irreducible is a division algebra), and this constraint underpins the orthogonality relations that make character theory work. Nearly every structural result in finite-group representation theory traces back to Schur's lemma.