Inverting the Frobenius map

The famous Frobenius characteristic map is a bijection from the space of characters of the symmetric group $S_n$ to the space of homogeneous symmetric functions of degree $n$. In this note, a formula for the inverse map is proved. More precisely, the generating function for the values of an arbitrary virtual character $\chi$ of $S_n$ is expressed in terms of the symmetric function which is the Frobenius image of $\chi$. We also give a $q$-analogue of this result by providing a similar formula for the Hecke algebra characters, and suggest some applications.