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.
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2$\begingroup$ certainly such conditions are not known in general, otherwise e.g. existence proofs for sporadic simple groups were much easier. For these groups, construction of the character table often was done before the construction of the group itself. $\endgroup$– Dima PasechnikCommented Mar 3, 2012 at 10:16
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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.(1-KNTS) 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.
answered Mar 2, 2012 at 23:31
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5$\begingroup$ Also relevant to give some historical perspective are papers by . Brauer (see MR0224725) and M.E. Harris (see MR0267012). A related topic which has received some attention a little more recently is that of "table algebras" ( see for example MR 1102572) $\endgroup$ Commented Mar 2, 2012 at 23:47