Frobenius-Schur functions
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The present paper is a detailed version of math/0003031. We introduce and study a new basis in the algebra of symmetric functions. The elements of this basis are called the Frobenius-Schur functions (FS-functions, 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. 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 FS-functions and the Schur functions. We also establish the FS analogs for a number of basic facts concerning the Schur 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 FS-functions and the ordinary Schur functions.
with appendix by Vladimir Ivanov. AMSTeX, 46 pages, 1 figure
with appendix by Vladimir Ivanov. AMSTeX, 46 pages, 1 figure