Tell me more ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.
up vote 3 down vote favorite

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.

share|improve this question

1 Answer

up vote 5 down vote accepted

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 $V$.

share|improve this answer
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"? – Alexei Averchenko Apr 10 '11 at 13:08
Well, yes, that's a tautology now. – wildildildlife Apr 10 '11 at 14:38
1  
@wild: why is that a tautology now? – Rasmus Apr 10 '11 at 14:47
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' :) – wildildildlife Apr 10 '11 at 18:43

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

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