Irreducible character as function of an idempotent

Let $𝑒 𝜇 𝛼 \in ℂ [𝐺]$ be a primitive idempotent generating the minimal left ideäl $𝐿 𝜇 𝛼$ carrying irrep $Γ 𝜇$ . Then the character $𝜒 𝜇$ of $Γ 𝜇$ is given by #m/thm/rep

𝜒 𝜇 (𝑔) = | 𝐶 (𝑔) | \sum ℎ \in [𝑔] \sim ―――― 𝑒 𝜇 𝛼 (ℎ)

where $[𝑔] \sim$ denotes the conjugacy class of $𝑔$ and $𝐶 (𝑔)$ its centraliser group with $| 𝐶 (𝑔) |$ equal to the size of the group divided by the size of the conjugacy class.

Proof

Using the inner product and convolution on $ℂ [𝐺]$

𝜒 𝜇 (𝑔 - 1) = \sum 𝑥 \in 𝐺 ⟨ 𝛿 𝑥 | Γ 𝜇 (𝑔 - 1) 𝛿 𝑥 ⟩ = \sum 𝑥 \in 𝐺 ⟨ 𝛿 𝑥 | Λ (𝑔 - 1) P ℂ [𝐺] (𝑒 𝜇 𝛼) 𝛿 𝑥 ⟩ = \sum 𝑥 \in 𝐺 ⟨ 𝛿 𝑥 | 𝛿 𝑔 - 1 𝑥 * 𝑒 𝜇 𝛼 ⟩ = \sum 𝑥, 𝑦 \in 𝐺 ―――― 𝛿 𝑥 (𝑦) [𝛿 𝑔 - 1 𝑦 * 𝑒 𝜇 𝛼] (𝑦) = \sum 𝑥 \in 𝐺 [𝛿 𝑔 - 1 𝑦 * 𝑒 𝜇 𝛼] (𝑥) = \sum 𝑥, 𝑦 \in 𝐺 𝛿 𝑔 - 1 𝑥 (𝑥 𝑦 - 1) 𝑒 𝜇 𝛼 (𝑦)

and since $𝑔 - 1 𝑥 = 𝑥 𝑦 - 1$ $⟹$ $𝑥 - 1 𝑔 - 1 𝑥 = 𝑦 - 1$ $⟹$ $𝑥 𝑔 𝑥 - 1 = 𝑦$ ,

𝜒 𝜇 (𝑔) = ――――― 𝜒 𝜇 (𝑔 - 1) = \sum 𝑥 \in 𝐺 ―――――― 𝑒 𝜇 𝛼 (𝑥 𝑔 𝑥 - 1)

Applying the Orbit-stabilizer theorem (see its proof), it follows that

𝜒 𝜇 (𝑔) = | 𝐶 (𝑔) | \sum ℎ \in [𝑔] \sim ―――― 𝑒 𝜇 𝛼 (ℎ)

as required.¹

It follows that $𝑑 𝜇 = | 𝐺 | ―――― 𝑒 𝜇 𝛼 (𝑒)$ .

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

An alternative proof is given in 2023, Groups and representations, pp. 60–61, but I like mine better. ↩

Irreducible character as function of an idempotent

Footnotes