Schur’s lemma

Irreducible representations of abelian groups are 1-dimensional

Let be an Abelian group. As a corollary of Schur’s lemma, if is a (complex) Irrep then it is a 1-dimensional irrep. rep Moreover, any abelian irrep is 1-dimensional.

Proof

If is abelian, then so are all its irreps. Let be an abelian irrep of on , so for every , and therefore is a multiple of the identity for all , so every subspace of is invariant under . Thus must be 1-dimensional in order for to be an irrep.

See also Main theorem.

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