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I just started reading a few lecture notes on representation theory and I was wondering about a big picture that one should keep in mind while reading through these lecture notes.

Have all (finite-dimensional) irreducible representations of finite groups and compact Lie groups over $\mathbb{C}$ been classified in some way?

I am guessing that all (finite-dimensional) irreducible representations of a general Lie group haven't yet been classified, or have they?

Thank you.

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    $\begingroup$ What do you mean by classified? Like for every finite group all its reps are known? Then no way. $\endgroup$
    – user641
    Aug 4, 2012 at 15:58
  • $\begingroup$ @SteveD I see. Thanks Steve! $\endgroup$
    – math-visitor
    Aug 4, 2012 at 17:17
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    $\begingroup$ A first problem with all finite groups is that those groups themselves have not been classified (only the simple ones have been, and that story is quite long enough as it stands). The usual way to get out of the implied difficulty (beacuse finite groups are Lie groups) when discussing Lie groups is to focus attention on connected Lie groups. The representation theory of connected compact or complex reductive matrix Lie groups is quite satisfactory. Here "reductive" means having no nontrivial normal subgroup consisting of unipotent matrices. Beyond that, life is harder. $\endgroup$
    – Marc van Leeuwen
    Aug 5, 2012 at 9:40
  • $\begingroup$ @MarcvanLeeuwen This is helpful; thank you. $\endgroup$
    – math-visitor
    Aug 5, 2012 at 17:18
  • $\begingroup$ @MarcvanLeeuwen you should your comment to be an answer. I think it would suffice $\endgroup$
    – Peter Patzt
    Mar 11, 2014 at 11:10

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A first problem with all finite groups is that those groups themselves have not been classified (only the simple ones have been, and that story is quite long enough as it stands). The usual way to get out of the implied difficulty (beacuse finite groups are Lie groups) when discussing Lie groups is to focus attention on connected Lie groups. The representation theory of connected compact or complex reductive matrix Lie groups is quite satisfactory. Here "reductive" means having no nontrivial normal subgroup consisting of unipotent matrices. Beyond that, life is harder

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