Idempotent of the complex group ring

Idempotent primitivity criterion

An idempotent 𝑒𝜇 ∈ℂ[𝐺] is primitive iff for every 𝑞 ∈ℂ[𝐺] there exists a scalar 𝜆𝑞 ∈ℂ such that 𝑒𝜇 ∗𝑞 ∗𝑒𝜇 =𝜆𝑞𝑒𝜇. rep

Proof

Let 𝐿𝜇𝛼 be the minimal left ideäl generated by primitive idempotent 𝑒𝜇𝛼. Since for all 𝑞,𝑠 ∈ℂ[𝐺]

Pℂ[𝐺](𝑒𝜇𝛼∗𝑞∗𝑒𝜇𝛼)Λℂ[𝐺](𝑠)=Λℂ[𝐺](𝑠)Pℂ[𝐺](𝑒𝜇𝛼∗𝑞∗𝑒𝜇𝛼)

and 𝐿𝜇𝛼 transforms in an irrep Γ𝜇, in particular

Pℂ[𝐺](𝑒𝜇𝛼∗𝑞∗𝑒𝜇𝛼)Γ𝜇𝛼(𝑔)=Γ𝜇𝛼(𝑔)Pℂ[𝐺](𝑒𝜇𝛼∗𝑞∗𝑒𝜇𝛼)

for all 𝑔 ∈𝐺 and thus by Schur’s lemma Pℂ[𝐺](𝑒𝜇𝛼 ∗𝑞 ∗𝑒𝜇𝛼) is 𝜆𝑞𝐈 in 𝐿𝜇𝛼 and zero everywhere else, i.e. 𝑒𝜇𝛼 ∗𝑞 ∗𝑒𝜇𝛼 =𝜆𝑞𝑒𝜇𝛼.

For the converse, assume 𝑒𝜇 is non-primitive and for every 𝑞 ∈ℂ[𝐺] there exists a scalar 𝜆𝑞 ∈ℂ such that 𝑒𝜇 ∗𝑞 ∗𝑒𝜇 =𝜆𝑞𝑒𝜇. From non-primitivity 𝑒𝜇 =𝑒1 +𝑒2 for nonzero idempotents with 𝑒1 ∗𝑒2 =0 =𝑒2 ∗𝑒1. Then on the one hand

𝑒𝜇∗𝑒1∗𝑒𝜇=(𝑒1+𝑒2)∗𝑒1∗(𝑒1+𝑒2)=𝑒1

but on the other hand

𝑒𝜇∗𝑒1∗𝑒𝜇=𝜆𝑒𝜇

so 𝜆𝑒𝜇 =𝑒1. But this is a contradiction, since it implies

𝑒1∗𝑒1=𝜆2𝑒𝜇∗𝑒𝜇=𝜆2𝑒𝜇=𝑒1=𝜆𝑒𝜇

and thus 𝜆 =𝜆2 and 𝜆 ≠0. Hence the assumption is false.¹

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tidy | en | SemBr

Footnotes

1. 2023, Groups and representations, p. 58 ↩