Schur’s lemma

Irreducible representations of abelian groups are 1-dimensional

Let 𝐺 be an Abelian group. As a corollary of Schur’s lemma, if Γ :𝐺 →GL(𝑉) is a (complex) Irrep then it is a 1-dimensional irrep. rep Moreover, any abelian irrep is 1-dimensional.

Proof

If 𝐺 is abelian, then so are all its irreps. Let Γ be an abelian irrep of 𝐺 on 𝑉, so Γ(ℎ)Γ(𝑔) =Γ(𝑔)Γ(ℎ) for every 𝑔,ℎ ∈𝐺, and therefore Γ(ℎ) is a multiple of the identity for all ℎ ∈𝐺, so every subspace of 𝑉 is invariant under 𝐺. Thus 𝑉 must be 1-dimensional in order for Γ to be an irrep.

See also Main theorem.

---

tidy | en | SemBr