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There is problem, asking, find all in-equivalent representations of an abelian group $G$.

My attempt: Let $f:G \to GL(V)$ a representation, by maschacke theorem $f_g$ is equivalent to direct sum of irreducible representations, $f^{(i)}_g$. Now as $G$ is abelian so, $f^{(i)}_g$'s are of degree one, so in matrix form $f_g$ diagonal matrix, rather exist $T$ such that $Tf_gT^{-1}$ is diagonal for all $g\in G$ where diagonal entries are clearly eigen values of $f_g$. Now what to do ? and what if $G$ is infinite ?

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  • $\begingroup$ possible duplicate of Complex finite dimensional irreducible representation of abelian group $\endgroup$
    – Dietrich Burde
    Jan 13, 2015 at 13:29
  • $\begingroup$ @DietrichBurde I would say it is not duplicated only because of the final part of the question, "what if $G$ is infinite?". $\endgroup$
    – Math137
    Jan 13, 2015 at 14:46
  • $\begingroup$ @math137: Well, I would say that the case where $G$ is infinite is also answered there. $\endgroup$
    – Dietrich Burde
    Jan 13, 2015 at 14:53
  • $\begingroup$ @DietrichBurde but that is not clear, at least to me $\endgroup$
    – Math137
    Jan 13, 2015 at 14:56
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    $\begingroup$ OK, I see your point. On the other hand, this has been discussed too, already - see the comments here. $\endgroup$
    – Dietrich Burde
    Jan 13, 2015 at 15:08

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