The representations of a finite group can be understood by their irreducible characters. A class function is a function from the group to the complex numbers that is constant on the conjugacy classes.
I know that any linear combination of the irreducible characters is the character of some representation. I also know that not all class functions are characters of a representation.
Let's say that I don't know all the irreducible characters of a group, but I come across a class function whose inner product with itself is 1. My question is: How do I know whether this function is actually the character of an irreducible representation?
More generally: How do I know whether a given class function is the character of some representation of a group without knowing all the irreducible representations?
EDIT: I see this question with answers: Class function as a character. This almost answers my question. To clarify what I am specifically interested in knowing, if I have found some irreducible representations of a group $G$. Say I have $\chi_1, \dots, \chi_m$. I know I haven't found all of them because I know the number of conjugacy classes. Then, say, I some other non-irreducible character $\chi$ and I know, say, that this is the character of some representation. Then I subtract a linear combination of $\chi_1, \dots, \chi_m$, and define the class function $\psi = \chi - (a_1\chi_1 + \dots + a_m\chi_m)$. How do I know whether this $\psi$ is the character of some representation?