Characterization of group characters I wonder how to characterize the characters of a (say, finite) group $G$ as special class functions, in particular for the case $G=S_n$ (symmetric group). The answer to this is presumably well known to people working in group theory, so it is more of a reference request.
Any character $\chi$ is "positive" in the sense that given $g_1,...,g_k\in G$, the matrix with coefficients $\chi(g_i^{-1}g_j)$ is positive. In case of $G=S_n$, characters are also known to be integer-valued class functions. Are there any further properties of characters that distinguish them from general class functions, or even a characterization of characters as class functions with a list of properties?
Of course, the characters are exactly those class functions which are linear combinations of the irreducible characters with non-negative integer coefficients, but that is not what I have in mind here.
 A: The best answer I can think of is Brauer's characterization of (generalized) characters: Recall that a generalized character is a difference of two characters. A Brauer elementary group is a group that is the direct product of a $p$-group and a cyclic group. Then Brauer's theorem states:  

A class function $\chi$ of a finite group $G$ is a generalized character if and only if its restriction $\chi_E$ to each Brauer elementary subgroup $E$ is a generalized character.  

This reduces the problem to certain subgroups of restricted structure. Since Brauer elementary groups are monomial, it is equivalent that $[\chi_E, \lambda ] \in \mathbb{Z}$ (inner product for class functions) for all linear characters $\lambda$ of all Brauer elementary subgroups $E\leq G$. (Thanks to Geoff Robinson for pointing this out in comment.)  
Unfortunately, the word "generalized" can not be omitted from the theorem. We get that a class function $\chi$ is an irreducible character if in addition $\chi(1)>0$ and $[\chi,\chi]=1$.
The result is treated, for example, in the books by Isaacs, by Huppert (Character Theory...), or by Serre (Linear Representations of Finite Groups).
