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In group theory, what is a complete representation and what is an incomplete representation?

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It probaly means one of the following. A complete representation of a group could be a faithful (1-1) one. I.E the group is isomorphic to its image. In this case, an incomplete representation would be a nonfaithful representation.

Another possibility is that you could call a representation of a proper subgroup $H<G$ an incomplete representation of G. That is to say, not all members of $G$ are represented in the image.

Just two words of advice:

  1. I'm, probaly wrong. I am a newcomer to representation theory.
  2. If you're reading a textbook that has these terms in them, the author should have included a definition of them.
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Thanks for your answer (and advice :p). I read it in a paper talking about the group $E_6$ and saying: a complete 27 representation of $E_6$. And the paper mentions another incomplete 27 rep, in the sense that not all members of the 27 are there. –  user09876 Jun 23 '12 at 21:01
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