4
$\begingroup$

What is meant by dimension of a representation in the following excersize: "Prove that any irreducible representation of an abelian group has the dimension of 1"? I looked at the solution, and it proves that any irreducible representation of an abelian group is scalar. I understand the proof, but I still can't figure out what is meant by dimension.

1 Answers 1

6

A representation of a group $G$ is a vector space $V$ together with an action of $G$ on $V$ by linear transformations. The dimension of the representation is just the dimension of $V$.

  • 0
    So should I interpret the exceresize as "Prove that an abelian group can only have irreducible representation on a vector space with dimension of 1"?2011-04-10
  • 0
    Well, yes, that's a tautology now.2011-04-10
  • 1
    @wild: why is that a tautology now?2011-04-10
  • 0
    I mean, given that Chris has just defined the dimension of a representation to be the dimension of the corresponding vector space, the exercise and your interpretation of it are tautologically the same. Anyway, the answer to your question is 'yes' :)2011-04-10