A combinatorial duality between the weak and strong Bruhat orders
A combinatorial duality between the weak and strong Bruhat orders
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弱 Bruhat 阶和强 Bruhat 阶之间的组合对偶性
DOI：
10.1016/j.jcta.2019.105178
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发表时间：
2018
期刊：
影响因子：
--
通讯作者：
Yibo Gao
中科院分区：
文献类型：
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作者：
Christian Gaetz;Yibo Gao
在最近的工作中，作者使用Stanley引入的降阶算子来证明对称群上弱Bruhat阶的强Sperner性质。Hamaker、Pechenik、Speyer和Weigandt将Schubert多项式的微分算子解释为Schubert多项式的微分算子，并利用这一点证明了Schubert多项式的一个新恒等式和Stanley的一个行列式猜想。本文研究了对称群上强Bruhat阶的提升算子Δ，它在许多方面都与λ是对偶的.我们证明了一个舒伯特恒等对偶的Hamaker等人。并推导出公式计算加权路径的Hasse图的强顺序同意路径计数公式的弱顺序，提供了一个强顺序模拟麦克唐纳的减少字的身份。我们还表明，功率的和Δ有相同的史密斯正规形式，我们明确描述，回答斯坦利的问题。
In recent work, the authors used an order lowering operator ∇, introduced by Stanley, to prove the strong Sperner property for the weak Bruhat order on the symmetric group. Hamaker, Pechenik, Speyer, and Weigandt interpreted ∇ as a differential operator on Schubert polynomials and used this to prove a new identity for Schubert polynomials and a determinant conjecture of Stanley. In this paper we study a raising operator Δ for thestrongBruhat order on the symmetric group, which is in many ways dual to ∇. We prove a Schubert identity dual to that of Hamaker et al. and derive formulas for counting weighted paths in the Hasse diagrams of the strong order which agree with path counting formulas for the weak order, providing a strong order analog of Macdonald's reduced word identity. We also show that powers of ∇ and Δ have the same Smith normal forms, which we describe explicitly, answering a question of Stanley.
舒伯特多项式的导数和斯坦利行列式猜想的证明
影响因子：
--
作者：
Hamaker, Zachary;Pechenik, Oliver;Speyer, David E;Weigandt, Anna
通讯作者：
Weigandt, Anna