Idempotent of the complex group ring

Idempotent primitivity criterion

An idempotent is primitive iff for every there exists a scalar such that . rep

Proof

Let be the minimal left ideäl generated by primitive idempotent . Since for all

and transforms in an irrep , in particular

for all and thus by Schur’s lemma is in and zero everywhere else, i.e. .

For the converse, assume is non-primitive and for every there exists a scalar such that . From non-primitivity for nonzero idempotents with . Then on the one hand

but on the other hand

so . But this is a contradiction, since it implies

and thus and . Hence the assumption is false.¹

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Footnotes

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