Abstract
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 (S-functions, 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 theFS-and S-functions. We also establish theFSanalogs for a number of basic facts concerning the S-functions: the Jacobi—Trudi formula together with its dual form, the combinatorial formula (expression in terms of tableaux), the Giambelli formula and the Sergeev—Pragacz formula.
All these results hold for a large family of bases interpolating between theFS-functions and the ordinary S-functions.
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Olshanski, G., Regev, A., Vershik, A., Ivanov, V. (2003). Frobenius-Schur Functions. In: Joseph, A., Melnikov, A., Rentschler, R. (eds) Studies in Memory of Issai Schur. Progress in Mathematics, vol 210. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0045-1_10
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