Given an $n\times n$ table of complex numbers, are there known sufficient conditions for the table to be the character table of a finite group? Representation theory gives plenty of necessary conditions, but I can't imagine they'd be enough in general.

You should look up an older article by Stephen Gagola, Jr., but read some of the arguments skeptically (as I did a long time ago when exploring this question in a graduate introduction to finite group representations): Gagola, Stephen M., Jr.(1KNTS) Formal character tables. Michigan Math. J. 33 (1986), no. 1, 3–10. I'm not sure whether more interesting results are known by now, but it's a difficult problem which has been around for a long time. I think it's fair to describe the problem as essentially open, though inevitable in this subject. 

