Research in Algebraic Combinatorics

该基金支持的研究领域是代数组合学，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入的见解，并解决组合学和其他领域的问题。组合学中研究的离散构型出现在数学、计算机科学、物理学、生物学和工程学的各个领域；DNA序列、系统发育树和通信网络都是离散配置的例子。组合方法在这些领域中发挥着越来越重要的作用。贯穿这个项目各个部分的一个共同主线是多项式的回文性和单模性。在代数、组合学和几何中出现的许多重要的枚举序列都是回文的和单峰的，但是证明单峰是相当具有挑战性的。单模性的证明出现在文献中，显著地使用了组合、解析、代数和代数几何技术。在PI目前的工作中，单模性问题导致了某些组合结构和几何结构之间有趣联系的发现，并发现了经典枚举公式的优雅的新q类似物。例如，广义欧拉多项式的回文性和单模性（由于De Mari， Procesi和Shayman）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，这些函数是Stanley的色对称函数的改进。本项目的目的之一是证明PI和Shareshian关于色拟对称函数与Tymoczko在a型正则半简单Hessenberg变化的上同调上的对称群表示之间关系的一个猜想。这可以建立Stanley和Stembridge关于色对称函数的一个长期的e正猜想。一个比回文性和单模性更强的性质，即伽马正性，目前在组合学和离散几何中引起了人们的兴趣，因为许多重要的枚举多项式类都是伽马正的。这个项目的一部分与最近与Dilks和Krattenthaler共同研究的伽马正性的q模拟有关。项目的另一部分继续PI关于偏置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的一个猜想，该猜想将加权分区的偏置集的某些传令集的拓扑与图关联面体的-正h多项式联系起来。

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.