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