0
$\begingroup$

Knowing the following theorem: " Every irreducible complex linear representation of an abelian group is one-dimensional " how can I prove the question mentioned in the title ..... Could anyone help me Please?

$\endgroup$

closed as off-topic by YiFan, Leucippus, Cesareo, Shailesh, Eevee Trainer Feb 20 at 0:59

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – YiFan, Leucippus, Cesareo, Shailesh, Eevee Trainer
If this question can be reworded to fit the rules in the help center, please edit the question.

1
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ but is this a proof? $\endgroup$ – Intuition Feb 20 at 13:47
  • $\begingroup$ will I include shur lemma in the proof? $\endgroup$ – Intuition Feb 20 at 13:49

Not the answer you're looking for? Browse other questions tagged or ask your own question.