ar X iv : m at h / 01 10 07 7 v 1 [ m at h . C O ] 7 O ct 2 00 1 FROBENIUS – SCHUR FUNCTIONS

We introduce and study a new basis in the algebra of symmetric functions. The elements of this basis are called the Frobenius–Schur functions (FSfunctions, for short). Our main motivation for studying the FS-functions is the fact that they enter a formula expressing the combinatorial dimension of a skew Young diagram in terms of the Frobenius coordinates. This formula plays a key role in the asymptotic character theory of the symmetric groups. The FS-functions are inhomogeneous, and their top homogeneous components coincide with the conventional Schur functions (Sfunctions, for short). The FS-functions are best described in the super realization of the algebra of symmetric functions. As supersymmetric functions, the FS-functions can be characterized as a solution to an interpolation problem. Our main result is a simple determinantal formula for the transition coefficients between the FSand S-functions. We also establish the FS analogs for a number of basic facts concerning the S-functions: Jacobi–Trudi formula together with its dual form; combinatorial formula (expression in terms of tableaux); Giambelli formula and the Sergeev–Pragacz formula. All these results hold for a large family of bases interpolating between the FSfunctions and the ordinary S-functions.