Equivalence of irreps on left ideals criterion

Let and be minimal left ideals transforming under the Regular group representation in irreps and respectively, and and be the generating primitive idempotents. Then iff for some . #m/thm/rep

Proof

If then there exists an intertwiner with and thus by lineärity

for all . Then has the required property, since

For the converse, let for some . Then

for all and in particular

for all , so by Schur's lemma the two irreps are equivalent.

Using lineärity arguments, it is sufficient to show for all to prove the idempotents generate non-equivalent irreps.

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