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.



Characterization of group characters
The best answer I can think of is **[Brauer's characterization of (generalized) characters](https://en.wikipedia.org/wiki/Brauer%27s_theorem_on_induced_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: 



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> 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. 
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> 
> 


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).


