Research in Algebraic Combinatorics

这笔拨款支持的研究是代数组合学，这是一个数学领域，旨在发展组合学(计算、排列和分析具体离散构型的科学)与涉及复杂抽象代数结构的纯数学领域之间的联系。我们的想法是利用这些联系来获得更深层次的见解，并解决组合学和其他领域的问题。组合数学中研究的离散构型出现在数学、计算机科学、物理、生物学和工程的各个领域；DNA序列、系统发育树和通信网络都是离散构型的例子。组合方法在这些领域发挥着越来越大的作用。贯穿本项目各个部分的一条共同的主线是多项式的回文和单峰。在代数、组合学和几何学中出现的许多重要的计数序列都是回文的和单峰的，但证明单峰可能是相当具有挑战性的。文献中出现的单峰性的证明惊人地利用了组合、解析、代数和代数几何技术。在PI目前的工作中，单峰性问题导致了某些组合和几何结构之间有趣的联系的发现，并发现了经典枚举式的优雅的新Q类似物。例如，广义欧拉多项式(由De Mari，Procesi和Shayman)的回文性质和单峰性在PI和Shareshian正在进行的关于色拟对称函数的工作中发挥了作用，这是Stanley色对称函数的改进。这个项目的一个目的是证明PI和Shareshian关于色拟对称函数与对称群的Tymoczko表示关于A型正则半单Hessenberg簇的上同调之间的关系的猜想。这可以建立一个由来已久的关于色对称函数的Stanley和Stembridge的e正性猜想。一个比回文性质和单峰性更强的性质，称为Gamma-正性，是当前组合学和离散几何中的一个重要性质，因为许多重要的可数多项式都是Gamma-正的。这个项目的一部分涉及最近与Dilks和Krtenthaler共同研究的伽马正性的Q模拟。该项目的另一部分继续PI关于偏序集拓扑和枚举组合之间相互作用的工作。一个目的是证明PI和她以前的学生Gonzalez D‘Leon将加权划分的偏序集的某些子集的拓扑连接到图的关联面体的Gamma正h-多项式的猜想。

The research supported by this grant is in algebraic combinatorics, which is an area of mathematics that seeks to develop connections between combinatorics (the science of counting, arranging, and analyzing concrete discrete configurations) and fields of pure mathematics that involve sophisticated abstract algebraic structures. The idea is to use these connections to gain deeper insights and solve problems in combinatorics and in the other fields. The discrete configurations that are studied in combinatorics arise in various fields of mathematics, computer science, physics, biology, and engineering; DNA sequences, phylogenetic trees, and communications networks are all examples of discrete configurations. Combinatorial methods are playing an increasing role in these fields.A common thread running through the various parts of this project is that of palindromicity and unimodality of polynomials. Many important enumerative sequences arising in algebra, combinatorics, and geometry are palindromic and unimodal, but proving unimodality can be quite challenging. Proofs of unimodality appearing in the literature have made striking use of combinatorial, analytic, algebraic, and algebro-geometric techniques. In the PI's current work, unimodality issues have led to the discovery of intriguing connections between certain combinatorial and geometric structures and to the discovery of elegant new q-analogs of classical enumerative formulas. For example, palindromicity and unimodality of a generalized Eulerian polynomial (due to De Mari, Procesi and Shayman) play a role in the PI's and Shareshian's ongoing work on chromatic quasisymmetric functions, which are a refinement of Stanley's chromatic symmetric functions. One aim of this project is to prove a conjecture of the PI and Shareshian on a relationship between the chromatic quasisymmetric functions and Tymoczko's representation of the symmetric group on the cohomology of the regular semisimple Hessenberg variety of type A. This could establish, among other things, a longstanding e-positivity conjecture of Stanley and Stembridge for chromatic symmetric functions. A property stronger than palindromicity and unimodality, known as gamma-positivity, is of current interest in combinatorics and discrete geometry, as many important classes of enumerative polynomials are gamma-positive. Part of this project is concerned with a q-analog of gamma-positivity recently studied in joint work with Dilks and Krattenthaler. Another part of the project continues the work of the PI on the interplay between poset topology and enumerative combinatorics. One aim is to prove a conjecture of the PI and her former student Gonzalez D'Leon connecting the topology of certain subposets of the poset of weighted partitions to gamma-positive h-polynomials of graph associahedra.