Geodesic
Inverting the Frobenius map
Algebra i analiz, Tome 10 (1998) no. 3, pp. 183-192.

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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. 
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     title = {Inverting the {Frobenius} map},
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S. Fomin; J.-Y. Thibon. Inverting the Frobenius map. Algebra i analiz, Tome 10 (1998) no. 3, pp. 183-192. http://geodesic.mathdoc.fr/item/AA_1998_10_3_a7/