We use k-Schur functions to get the minimal boundary of the k-bounded partition poset. This permits to describe the central random walks on affine Grassmannian elements of type A and yields a polynomial expression for their drift. We also recover Rietsch's parametriza-tion of totally nonnegative unitriangular Toeplitz matrices without using quantum cohomology of flag varieties. All the homeomorphisms we define can moreover be made explicit by using the combinatorics of k-Schur functions and elementary computations based on Perron-Frobenius theorem.
From symmetric fundamental expansions to Schur positivity
We consider families of quasisymmetric functions with the property that if a symmetric function $f$ is a positive sum of functions in one of these families, then f is necessarily a positive sum of Schur functions. Furthermore, in each of the families studied, we give a combinatorial description of the Schur coefficients of $f$ . We organize six such families into a poset, where functions in higher families in the poset are always positive integer sums of functions in each of the lower families. This poset includes the Schur functions, the quasisymmetric Schur functions, the fundamental quasisy…
Analogs of Schur functions for rank two Weyl groups obtained from grid-like posets
在先前的工作中，作者，以及M。麦克拉德河A. Proctor和N. J. Wildberger研究了与秩2根系/外尔群相关的“外尔双交替”（又称“外尔特征”）的某些分配格模型。这些分配格被统一地描述为从某些格状偏序集中取来的阶理想的格，尽管将格与Weyl双交错格连接起来的参数是根据秩2根系的类型而逐个情况的。利用这种与Weyl双交替的联系，这些格被证明是秩对称的和秩单峰的，并且它们的秩生成函数被证明具有美丽的乘积的表达式。在这里，这些结果是重新从头开始使用完全统一的和基本的组合推理，结合一些组合方法，在其他地方开发的第二个上市的作者。
最近引入了一种称为k形的偏序集，为a（k-1）-Schur函数在t=1时扩展为k-Schur函数提供了一个组合规则。这个构造的主要成分是一个双射，我们称之为弱双射，它将一个由（k-1）-表和k-形偏序集中的一条路组成的对与一个k-表联系起来。我们在这里定义了k-tableaux上的电荷概念（推测它给出了Hall-Littlewood多项式到k-Schur函数的展开系数的组合解释），并表明它在标准情况下与弱双射兼容。特别地，我们得到了一个标准的尺寸为n的tableau的通常电荷等于它在k-形偏序集中对应的路的电荷之和，其中k= 2，3. n.
Skew quasisymmetric Schur functions and noncommutative Schur functions
Recently a new basis for the Hopf algebra of quasisymmetric functions $QSym$ , called quasisymmetric Schur functions, has been introduced by Haglund, Luoto, Mason, van Willigenburg. In this paper we extend the definition of quasisymmetric Schur functions to introduce skew quasisymmetric Schur functions. These functions include both classical skew Schur functions and quasisymmetric Schur functions as examples, and give rise to a new poset $\mathcal{L}_C$ that is analogous to Young's lattice. We also introduce a new basis for the Hopf algebra of noncommutative symmetric functions $NSym$ . This b…
Symmetries on the lattice of $k$-bounded partitions
2002年R.苏特确定了二面角对称的某些阶理想在杨格，并给出了一个组合行动的分区在这些阶理想。从几何上看，在仿射对称群的几何实现中，阶理想可以看作是与凹室在2倍膨胀中的双射。通过考虑m重伸缩，我们观察到L所考虑的k-有界分划格中的一个更大的阶理想集。Lapointe，A. Lascoux，and J. Morse在k-Schur函数的研究中。我们确定的阶理想和它的循环作用明确的几何和组合的形式。
