Character irreducibility criterion

Let $Γ : 𝐺 \to G L (𝑉)$ be a complex Group representation and $𝜒 : 𝐺 \to ℂ$ be the corresponding Group character. Then $Γ$ is reducible iff #m/thm/rep

1 | 𝐺 | \sum 𝑔 \in 𝐺 | 𝜒 (𝑔) | 2 = 1

and otherwise the sum is $> 1$ .

Proof

Let $Γ : 𝐺 \to 𝖵 𝖾 𝖼 𝗍 ℂ$ be a (in general reducible) representation with

Γ ≅ 𝑚 ⨁ 𝑗 = 1 𝑎 𝑗 ⨁ 𝑘 = 1 Γ 𝑗

i.e. each irrep $Γ 𝑗$ occurs $𝑎 𝑗$ times. Then it follows from the definition of a character as a trace that

𝜒 (𝑔) = 𝑚 \sum 𝑗 𝑎 𝑗 𝜒 𝑗 (𝑔)

\sum 𝑔 \in 𝐺 ―――― 𝜒 𝑗 (𝑔) 𝜒 𝑘 (𝑔) = | 𝐺 | 𝛿 𝑗 𝑘

it follows that

1 | 𝐺 | \sum 𝑔 \in 𝐺 | 𝜒 (𝑔) | 2 = \sum 𝑗, 𝑘 𝑎 𝑗 𝑎 𝑘 𝛿 𝑗 𝑘 = \sum 𝑗 (𝑎 𝑗) 2

as required.

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