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