Abelian representation

An abelian representation $Γ : 𝐺 \to G L (𝑉)$ is a representation whose range $Γ (𝐺) \subseteq G L (𝑉)$ is abelian, #m/def/rep i.e. for all $𝑔, ℎ \in 𝐺$

Γ (𝑔 ℎ) = Γ (𝑔) Γ (ℎ) = Γ (ℎ) Γ (𝑔) = Γ (ℎ 𝑔)

Main theorem

A representation $Γ : 𝐺 \to G L (𝑉)$ is abelian iff it is the direct sum of 1-dimensional irrep. #m/thm/rep

Proof

If $Γ ≅ ⨁ 𝑘 \in ˆ 𝐴 𝑎 𝑘 𝜒 𝑘$ then it is immediately abelian since there exists a reälization in which matrices are simultaneously diagonal, and hence commute. Conversely, Let $𝑉 = ⨁ 𝑉 𝑖$ be the decomposition of $𝑉$ into irreducible invariant subspaces. Since $Γ$ is abelian in each of these subspaces, the irrep carried thereby they must be 1-dimensional. Hence $Γ$ is the direct sum of 1-dimensional irreps.

#state/tidy | #lang/en | #SemBr