If $G$ is an abelian group, then every irreducible character has dimension one (i.e. is linear), for finite group we also have a converse. Do we have a converse for infinite groups? Or:

Does there exists an infinite group all whose irreducible characters are linear, which is not abelian?

I append a proof that $G$ abelian implies all irreducible characters are linear: If $G$ is an abelian group and $V$ an irreduble $\mathbb C[G]$-module, then by Schur's lemma we have $\operatorname{dim}\operatorname{Hom}_{\mathbb C[G]}(V, V) = 1$. As $G$ is abelian, every $g$ corresponds to a $\mathbb C[G]$-homomorphism. Hence in its action on $V$ could be identified with some homomorphism of the form $\lambda \cdot I_n$ with $n = \operatorname{dim}V$. If $n > 1$ then such a map would leave every space invariant, also the ones of dimension $< n$, hence $n = 1$. $\square$

  • $\begingroup$ For the form of Schur's lemma you're using you need the assumption that $V$ is finite-dimensional. $\endgroup$ – Qiaochu Yuan Jan 17 '16 at 23:06
  • $\begingroup$ @QiaochuYuan Thanks for pointing out! So this direction is also not working in general? $\endgroup$ – StefanH Jan 17 '16 at 23:06
  • $\begingroup$ Well, it depends on whether you care about irreducible characters or irreducible representations. If the latter, sometimes irreducible representations are infinite-dimensional, so don't have characters in any obvious sense. $\endgroup$ – Qiaochu Yuan Jan 17 '16 at 23:07
  • $\begingroup$ @QiaochuYuan Okay, think to answer that satisfactory I need to learn more representation theory, I just started, so maybe I will come back for more specific questions later when I understand the theory better. $\endgroup$ – StefanH Jan 17 '16 at 23:15

Yes. Take an infinite simple group $G$, such as $PSL_3(K)$ for a sufficiently large field $K$ of sufficiently large characteristic, with cardinality strictly larger than $\mathbb{C}$. Then every finite-dimensional representation of $G$ over $\mathbb{C}$ is trivial, so its only irreducible character is the trivial one.

There are more complicated examples of finitely generated or even finitely presented groups all of whose finite-dimensional representations are trivial.

  • $\begingroup$ Really abstract example. Do you have any references where to read more about this? $\endgroup$ – StefanH Jan 17 '16 at 22:03
  • $\begingroup$ Not really. This question comes up on MO and math.SE periodically. Examples are necessarily a bit strange because most familiar groups are linear, meaning that they have faithful finite-dimensional representations. $\endgroup$ – Qiaochu Yuan Jan 17 '16 at 23:03
  • $\begingroup$ I also disagree that this is a really abstract example. The projective special linear groups are relatively concrete as far as groups go. A lot is known about them, and you can represent their elements explicitly using matrices. That's a lot better than weirder examples like the finitely generated groups with no nontrivial finite-dimensional reps. $\endgroup$ – Qiaochu Yuan Jan 17 '16 at 23:05
  • $\begingroup$ Okay, thanks for the comments! $\endgroup$ – StefanH Jan 17 '16 at 23:06

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