INVERTING THE FROBENIUS MAP

Abstract. The famous Frobenius characteristic map is a bijection from the space of characters of the symmetric group S „ 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 x of Sn is expressed in terms of the symmetric function which is the Frobenius image of x-We also give a ^-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, Л = A(xi,x-2,...) will denote the algebra of symmetric functions (i.e., symmetric formal power series of bounded degree) in infinitely many variables The celebrated Frobenius correspondence is a linear isomorphism between the space of class functions on (the set conjugacy classes of) the symmetric group Sn and the space Л " of homogeneous symmetric functions of degree n. Specifically, it sends each irreducible character xA to the corresponding Schur function s\. Since the conjugacy classes of S „ are labeled by partitions of n, with any class function x o n $n w e can associate a symmetric generating function] T X(a)ma; (1.1) here ma denotes the monomial symmetric function, and x ( a) is the value of x on the elements of the conjugacy class labeled by a. Identifying every x with the corresponding generating function (1.1), we can view the Frobenius map as the linear automorphism of Л " given by ^2xX(a)ma i—> sx