Bruhat and balanced graphs, manifolds, partitions and affine permutations

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。该提案由代数，几何和拓扑组合学的五个广泛的研究课题组成。 第一个项目是将Coxeter系统的强Bruhat图的结果扩展到更一般的平衡图。 PI最近将Billera和Brenti关于Bruhat图的cd指数的工作扩展到平衡图，并显示了Kazhdan-Lusztig多项式，一个重要的拓扑和表示理论不变量，也扩展到这个设置。 第二个计画是利用cd指标研究流形的正规细分的面关联资料。 PI将研究凸多面体的经典结果的流形类似物，包括确定cd指数系数的不等式，以Zaslavsky关于超平面安排的开创性工作的精神发展正则流形安排的理论，并将Stanley的球形壳性概念扩展到流形。受Wachs关于d-可分划分格的工作的启发，第三个项目是从偏序集同调、可壳性和表示论的角度研究限制划分格;第四个项目是第二类斯特林数和Bell数的Dowling类似数的经典计数研究。 第五个项目是研究仿射置换群的组合置换统计，如超数、降数和主指数。组合数学本质上是一个跨学科的研究领域，连接了数学和科学的许多领域。 这一提议进一步扩展了组合数学的范围。 例如，Kazhdan-Lusztig多项式是最初在拓扑学中定义的深度不变量。 在这个项目中，他们将从组合的角度进行分析，以进一步提高我们对他们的理解。 适用于多面体的理论，这是球形的对象，将被扩展到其他流形，具有更复杂的拓扑结构。 发展我们的拓扑观点很重要，因为现代物理学的很大一部分都集中在研究空间的拓扑结构上。 加深我们对排列统计和基本组合枚举的理解可能有助于我们分析和识别大量基因组数据中的模式，以及改善包括互联网在内的通信。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).This proposal is composed of five broad research topics in algebraic, geometric and topological combinatorics. The first project is to expand results about the strong Bruhat graph of Coxeter systems to the more general setting of balanced graphs. The PI has recently extended Billera and Brenti's work on the cd-index of Bruhat graphs to balanced graphs and showed the Kazhdan-Lusztig polynomials, an important topological and representation theoretic invariant, also extend to this setting. The second project is a study of the face incidence data of regular subdivisions of manifolds via the cd-index. The PI will investigate manifold analogues of classical results for convex polytopes, including to determine inequalities for the coefficients of the cd-index, to develop a theory of regular manifold arrangements in the spirit of Zaslavsky's seminal work on hyperplane arrangements, and to extend Stanley's notion of spherical shellability to manifolds. Motivated by Wachs' work on the d-divisible partition lattice, the third project is to examine the restricted partition lattice from the standpoint of poset homology, shellability and representation theory.The fourth project is a classical enumerative study of Dowling analogues of the Stirling numbers of the second kind and the Bell numbers. The fifth project is to investigate combinatorial permutation statistics, such as excedances, descents and the major index, for the group of affine permutations.Combinatorics is inherently an interdisciplinary field of study linking many areas of mathematics and the sciences. This proposal further expands the range of combinatorics. For example, the Kazhdan-Lusztig polynomials are a deep invariant originally defined in topology. In this project they will be analyzed from a combinatorial perspective to further enhance our understanding of them. Theories that apply to polytopes, which are sphere-like objects, will be extended to other manifolds which have more complicated topological structure. Developing our topological perspective is important since a large part of modern-day physics is focused on studying the topology of space. Deepening our understanding of permutation statistics and basic combinatorial enumeration may help us to analyze and recognize patterns in vast genome data, as well as to improve communications, including the internet.