Mathematics – Combinatorics
Scientific paper
2001-10-07
In: Studies in memory of Issai Schur (Chevaleret/Rehovot, 2000), Progr. Math., vol. 210, Birkhauser Boston, 2003, 251--299
Mathematics
Combinatorics
with appendix by Vladimir Ivanov. AMSTeX, 46 pages, 1 figure
Scientific paper
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.
Olshanski Grigori
Regev Amitai
Vershik Anatoly
No associations
LandOfFree
Frobenius-Schur functions does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Frobenius-Schur functions, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Frobenius-Schur functions will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-486133