We give a direct proof of the equivalence between the Giambelli and Pieri type formulas for Hall-Littlewood functions using Young's raising operators, parallel to joint work with Buch and Kresch for the Schubert classes on isotropic Grassmannians. We prove several closely related mirror identities enjoyed by the Giambelli polynomials, which lead to new recursions for Schubert classes. The raising operator approach is applied to obtain tableau formulas for Hall-Littlewood functions, theta polynomials, and related Stanley symmetric functions. Finally, we introduce the notion of a skew element w of the hyperoctahedral group and identify the set of reduced words for w with the set of standard k-tableaux on a skew Young diagram.
Backward jeu de taquin slides for composition tableaux and a noncommutative Pieri rule
我们给出了一个半标准的反向合成表上的反向jeu de taquin幻灯片模拟。这些tableaux首先由Haglund，Luoto，Mason和van Willigenburg在定义拟对称Schur函数时研究。我们的算法执行向后jeu德taquin幻灯片半标准的反向组成tableaux结果在一个自然的操作组合，我们称之为jdt操作。这个操作员反过来又产生了一个新的偏序集结构的组成，我们列举的最大链。作为应用，我们还给出了一个非交换的Pieri规则的非交换Schur函数，使用的jdt运营商。
We introduce and study a family of inhomogeneous symmetric functions which we call the Frobenius-Schur functions. These functions are indexed by partitions and differ from the conventional Schur functions in lower terms only. Our interest in these new functions comes from the fact that they provide an explicit expression for the dimension of a skew Young diagram in terms of the Frobenius coordinates. This is important for the asymptotic theory of the characters of the symmetric groups . Our main result is a surprisingly simple determinantal expression of the Frobenius-Schur functions in terms …
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 co…
