Schubert polynomials, slide polynomials, Stanley symmetric functions and quasi-Yamanouchi pipe dreams
Schubert polynomials, slide polynomials, Stanley symmetric functions and quasi-Yamanouchi pipe dreams
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DOI：
10.1016/j.aim.2016.10.015
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发表时间：
2016-03
影响因子：
1.7
通讯作者：
Sami H. Assaf;D. Searles
中科院分区：
文献类型：
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作者：
Sami H. Assaf;D. Searles
我们引入了两个新的基地多项式，解除单项式和基本拟对称函数的完整的多项式环。通过定义一个新的条件上的梦想，所谓的准山之内，我们给出了一个积极的组合规则扩展舒伯特多项式到这些新的基地，平行的扩张Schur函数到基本的准对称函数。结果，我们得到了Schubert多项式到Stanley对称函数的稳定极限的一个改进。我们还给出了这些基的正结构常数的组合规则，分别推广了拟混洗积和混洗积。我们用它来给一个Littlewood-Richardson规则扩展的产品舒伯特多项式到基本的滑动多项式，并给出公式的产品斯坦利对称函数的舒伯特结构常数。
We introduce two new bases for polynomials that lift monomial and fundamental quasisymmetric functions to the full polynomial ring. By defining a new condition on pipe dreams, called quasi-Yamanouchi, we give a positive combinatorial rule for expanding Schubert polynomials into these new bases that parallels the expansion of Schur functions into fundamental quasisymmetric functions. As a result, we obtain a refinement of the stable limits of Schubert polynomials to Stanley symmetric functions. We also give combinatorial rules for the positive structure constants of these bases that generalize the quasi-shuffle product and shuffle product, respectively. We use this to give a Littlewood–Richardson rule for expanding a product of Schubert polynomials into fundamental slide polynomials and to give formulas for products of Stanley symmetric functions in terms of Schubert structure constants.