Ideal of the complex group ring

Trivial irrep carrying ideäl of the group ring

The primitive idempotent

𝑒1=1|𝐺|∑𝑔∈𝐺𝛿𝑔

generates the one-dimensional left ideäl 𝐿1 carrying the trivial irrep.

Λ(𝑔)𝑞∗𝑒1=𝛿𝑔∗∑𝑥∈𝐺𝑞(𝑥)𝛿𝑥∗𝑒1=1|𝐺|𝛿𝑔∗∑𝑥,𝑦∈𝐺𝑞(𝑥)𝛿𝑦=1|𝐺|∑𝑥,𝑦∈𝐺𝑞(𝑥)𝛿𝑔𝑥𝑦=1|𝐺|∑𝑥,𝑧∈𝐺𝑞(𝑥)𝛿𝑥𝑧=∑𝑥∈𝐺𝑞(𝑥)𝛿𝑥∗𝑒1=𝑞∗𝑒1

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