Idempotent of the complex group ring

Irreducible character as function of an idempotent

Let 𝑒𝜇𝛼 ∈ℂ[𝐺] be a primitive idempotent generating the minimal left ideäl 𝐿𝜇𝛼 carrying irrep Γ𝜇. Then the character 𝜒𝜇 of Γ𝜇 is given by rep

𝜒𝜇(𝑔)=|𝐶(𝑔)|∑ℎ∈[𝑔]∼――――𝑒𝜇𝛼(ℎ)

where [𝑔]∼ 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]

𝜒𝜇(𝑔−1)=∑𝑥∈𝐺⟨𝛿𝑥|Γ𝜇(𝑔−1)𝛿𝑥⟩=∑𝑥∈𝐺⟨𝛿𝑥|Λ(𝑔−1)Pℂ[𝐺](𝑒𝜇𝛼)𝛿𝑥⟩=∑𝑥∈𝐺⟨𝛿𝑥|𝛿𝑔−1𝑥∗𝑒𝜇𝛼⟩=∑𝑥,𝑦∈𝐺――――𝛿𝑥(𝑦)[𝛿𝑔−1𝑦∗𝑒𝜇𝛼](𝑦)=∑𝑥∈𝐺[𝛿𝑔−1𝑦∗𝑒𝜇𝛼](𝑥)=∑𝑥,𝑦∈𝐺𝛿𝑔−1𝑥(𝑥𝑦−1)𝑒𝜇𝛼(𝑦)

and since 𝑔−1𝑥 =𝑥𝑦−1 ⟹ 𝑥−1𝑔−1𝑥 =𝑦−1 ⟹ 𝑥𝑔𝑥−1 =𝑦,

𝜒𝜇(𝑔)=―――――𝜒𝜇(𝑔−1)=∑𝑥∈𝐺――――――𝑒𝜇𝛼(𝑥𝑔𝑥−1)

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

𝜒𝜇(𝑔)=|𝐶(𝑔)|∑ℎ∈[𝑔]∼――――𝑒𝜇𝛼(ℎ)

as required.¹

It follows that 𝑑𝜇 =|𝐺|――――𝑒𝜇𝛼(𝑒).

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Footnotes

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