Affine dual equivalence and $k$-Schur functions
Affine dual equivalence and $k$-Schur functions
复制标题
仿射对偶等价和 $k$-Schur 函数
DOI：
10.4310/joc.2012.v3.n3.a5
复制
发表时间：
2012
期刊：
影响因子：
--
通讯作者：
Sara C. Billey
中科院分区：
文献类型：
--
作者：
Sami H. Assaf;Sara C. Billey
k-Schur函数首先由Lapointe、Lascoux和莫尔斯[18]引入，希望将麦克唐纳多项式的展开式细化为Schur函数。最近，Lam，Lapointe，莫尔斯和Shimozono [17]给出了k-Schur函数的另一种定义，作为与仿射对称群模对称群上的Bruhat阶的标记饱和链对应的星号强tableaux的加权生成函数。这个定义已经被证明对应于A型仿射格拉斯曼的Schubert基[15]，并且在t = 1时，它等价于Lapointe和莫尔斯[22]的k-表表征。本文将Haiman关于标准Young表[12]的对偶等价关系推广到全星强表。基本等价关系可以被解释为图中的标记边，它们具有Assaf对偶等价图的许多性质。这些图显示了使用k-舒尔函数和Sn/Sn上的区间结构的复杂性。我们引进的概念，扁平化和挤压斜星强tableaux在类比jeu德taquin幻灯片，以给出一种方法来找到所有同构类型的仿射对偶等价图的秩4。最后，我们指出了在推广对偶等价的其他方法中存在的一些问题。
The k-Schur functions were first introduced by Lapointe, Lascoux and Morse [18] in the hopes of refining the expansion of Macdonald polynomials into Schur functions. Recently, an alternative definition for k-Schur functions was given by Lam, Lapointe, Morse, and Shimozono [17] as the weighted generating function of starred strong tableaux which correspond with labeled saturated chains in the Bruhat order on the affine symmetric group modulo the symmetric group. This definition has been shown to correspond to the Schubert basis for the affine Grassmannian of type A [15] and at t = 1 it is equivalent to the k-tableaux characterization of Lapointe and Morse [22]. In this paper, we extend Haiman’s dual equivalence relation on standard Young tableaux [12] to all starred strong tableaux. The elementary equivalence relations can be interpreted as labeled edges in a graph which share many of the properties of Assaf’s dual equivalence graphs. These graphs display much of the complexity of working with k-Schur functions and the interval structure on Sn/Sn. We introduce the notions of flattening and squashing skew starred strong tableaux in analogy with jeu de taquin slides in order to give a method to find all isomorphism types for affine dual equivalence graphs of rank 4. Finally, we state some open problems on other ways to generalize dual equivalence.