Coxeter群中的弱Bruhat链

The weak Bruhat order on Coxeter groups, closely related with Schubert polynomials and Stanley symmetric functions, is a hot topic in both combinatorics and algebraic geometry. This project mainly focuses on the following two questions about the maximal weak Bruhat chains in Coxeter groups: .(1) Find the immanent cause of the coincidence of the weighted sum of the maximal weak Bruhat chains and that of maximal strong Bruhat chains in the symmetric group. Considering the difference between the two kinds of Bruhat chains, the interrelation between them will stimulate the integration of the theory and research tools on the two kinds of Bruhat chains..(2) Compute the diameter of the graph with the maximal weak Bruhat chains its vertices and Coxeter-Knuth relations its edges. Coxeter-Knuth relation is a mixture of the Coxeter relation and the Knuth relation, which makes it have both the geometric properties of the Coxeter relation and the algorithmic properties of Knuth relation. The study of Coxeter-Knuth relation in this project will promote the application of the algorithmic tools in geometry.

Coxeter群上的弱Bruhat序，与Schubert多项式和Stanley对称多项式紧密相关，是组合学和代数几何中的热点课题。本项目主要研究关于Coxeter群上极长弱Bruhat链的两方面问题：.（1）寻找对称群中极长弱Bruhat链和极长强Bruhat链权重之和相等的内在原因。鉴于两类Bruhat链间的较大差异，寻求二者间的内在联系将在一定程度上刺激两类Bruhat链的理论和研究工具的融合。.（2）计算以极长弱Bruhat链为顶点，以Coxeter-Knuth关系为边的图的直径。Coxeter-Knuth关系，作为Coxeter关系和Knuth关系的混合体，兼具Coxeter关系的几何性和Knuth关系算法方面的特点。本项目关于Coxeter-Knuth关系的研究将会促进算法工具在几何中的应用。

Bruhat序和弱Bruhat序是代数几何、表示论和组合数学中的重要研究对象。本项目的主要目的就是探究两类Bruhat序的组合性质，及其在格路等中的应用。通过对两类Bruhat序的研究，项目组成员发现这两类Bruhat序的性质可以用来构造自然的对合，组合学中很多行列式的组合意义都可以由此给出。特别地，项目主持人与合作者利用这种对合方式证明了Morales、Pak和Panova关于反平面分拆生成函数的猜想。