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Prove that every irreducible real representation of an abelian group is one or two dimensional.

I know the following theorem (a consequence of Schur's lemma): "Every irreducible complex linear representation of an abelian group is one-dimensional". How can I use this to prove the above?

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Take an irreducible complex representation (of dimension 1) and think of it as a real representation (of dimension 2). Clearly it either decomposes as either a sum of two one dimensional irreducibles or it is a two dimensional irreducible real representation.

But now I claim that every real representation arises this way: We know that every representation of $G$ over $\mathbb{R}$ or $\mathbb{C}$ appears inside the corresponding regular representation $\mathbb{R}[G]$ or $\mathbb{C}[G]$, but as a real representation $\mathbb{C}[G] = \mathbb{R}[G] + i\mathbb{R}[G]$ so every irreducible real representation appears inside the restriction of an irreducible complex representation.

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  • $\begingroup$ but is this a proof? $\endgroup$
    – Emptymind
    Feb 20, 2019 at 13:47
  • $\begingroup$ will I include shur lemma in the proof? $\endgroup$
    – Emptymind
    Feb 20, 2019 at 13:49
  • $\begingroup$ How do we conclude in the end that every irreducible real representation appears as restriction of an irreducible complex representation. Can you please explain. $\endgroup$
    – Nancy
    May 28, 2022 at 18:16

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