Idempotent of the complex group ring

Idempotent primitivity criterion

An idempotent π‘’πœ‡ βˆˆβ„‚[𝐺] is primitive iff for every π‘ž βˆˆβ„‚[𝐺] there exists a scalar πœ†π‘ž βˆˆβ„‚ such that π‘’πœ‡ βˆ—π‘ž βˆ—π‘’πœ‡ =πœ†π‘žπ‘’πœ‡. #m/thm/rep

Proof

Let πΏπœ‡π›Ό be the minimal left ideΓ€l generated by primitive idempotent π‘’πœ‡π›Ό. Since for all π‘ž,𝑠 βˆˆβ„‚[𝐺]

Pβ„‚[𝐺](π‘’πœ‡π›Όβˆ—π‘žβˆ—π‘’πœ‡π›Ό)Ξ›β„‚[𝐺](𝑠)=Ξ›β„‚[𝐺](𝑠)Pβ„‚[𝐺](π‘’πœ‡π›Όβˆ—π‘žβˆ—π‘’πœ‡π›Ό)

and πΏπœ‡π›Ό transforms in an irrep Ξ“πœ‡, in particular

Pβ„‚[𝐺](π‘’πœ‡π›Όβˆ—π‘žβˆ—π‘’πœ‡π›Ό)Ξ“πœ‡π›Ό(𝑔)=Ξ“πœ‡π›Ό(𝑔)Pβ„‚[𝐺](π‘’πœ‡π›Όβˆ—π‘žβˆ—π‘’πœ‡π›Ό)

for all 𝑔 ∈𝐺 and thus by Schur's lemma Pβ„‚[𝐺](π‘’πœ‡π›Ό βˆ—π‘ž βˆ—π‘’πœ‡π›Ό) is πœ†π‘žπˆ in πΏπœ‡π›Ό and zero everywhere else, i.e. π‘’πœ‡π›Ό βˆ—π‘ž βˆ—π‘’πœ‡π›Ό =πœ†π‘žπ‘’πœ‡π›Ό.

For the converse, assume π‘’πœ‡ is non-primitive and for every π‘ž βˆˆβ„‚[𝐺] there exists a scalar πœ†π‘ž βˆˆβ„‚ such that π‘’πœ‡ βˆ—π‘ž βˆ—π‘’πœ‡ =πœ†π‘žπ‘’πœ‡. From non-primitivity π‘’πœ‡ =𝑒1 +𝑒2 for nonzero idempotents with 𝑒1 βˆ—π‘’2 =0 =𝑒2 βˆ—π‘’1. Then on the one hand

π‘’πœ‡βˆ—π‘’1βˆ—π‘’πœ‡=(𝑒1+𝑒2)βˆ—π‘’1βˆ—(𝑒1+𝑒2)=𝑒1

but on the other hand

π‘’πœ‡βˆ—π‘’1βˆ—π‘’πœ‡=πœ†π‘’πœ‡

so πœ†π‘’πœ‡ =𝑒1. But this is a contradiction, since it implies

𝑒1βˆ—π‘’1=πœ†2π‘’πœ‡βˆ—π‘’πœ‡=πœ†2π‘’πœ‡=𝑒1=πœ†π‘’πœ‡

and thus πœ† =πœ†2 and πœ† β‰ 0. Hence the assumption is false.1

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

Footnotes

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