Inverting the Frobenius Map

The famous Frobenius characteristic map is a bijection from the space of characters of a symmetric group S n to the space of homogeneous symmetric functions of degree n. In this note, we prove a formula for the inverse map. More precisely, we express the generating function for the values of an arbitrary virtual character of S n in terms of the symmetric function which is the Frobenius image of. We also give a q-analogue of this result by providing a similar formula for the Hecke algebra characters, and suggest some applications. 1. The symmetric group In this note, we follow the terminology and notation of Macdonald M]. In particular, = (x 1 ; x 2 ; : : :) will denote the algebra of symmetric functions (i.e., symmetric formal power series of bounded degree) in innnitely many variables x 1 ; x 2 ; : : :. The celebrated Frobenius correspondence is a linear isomorphism between the space of class functions on (the set conjugacy classes of) the symmetric group S n , on one side, and the space n of homogeneous symmetric functions of degree n, on another. Speciically, it sends each irreducible character to the corresponding Schur function s .