Research in Algebraic Combinatorics

该项目的主要目标是继续与John Shareshian共同研究对称函数理论、枚举组合学和代数几何之间的相互作用。该项目源于对一类对称函数的研究，这些函数出现在各种情况下，例如Carlitz， Scoville和Vaughan关于Smirnov词枚举的工作，Procesi和Stanley关于与a型根系统相关的环面变异的上同调的对称群表示的工作，以及研究者和Shareshian关于欧拉多项式的q-模拟的工作。为了理解这些结构之间关系的基础，研究者和Shareshian对这种关系提出了一个深远的概括，本项目将对此进行探索。推测的推广涉及Stanley的色对称函数的准对称改进，Tymoczko在a型的正则半简单Hessenberg变分的上同调上的对称群的表示，以及欧拉多项式的推广（由于De Mari， Procesi和Shayman）的q-模拟。这种推测关系将提供一种代数-几何方法来攻击Stanley和Stembridge关于色对称函数的长期存在的$e$-正猜想，以及研究者和Shareshian关于其q-类似De Mari-Procesi-Shayman广义欧拉多项式的单模猜想。该项目还将继续研究偏序集拓扑和枚举组合之间的相互作用。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。

The main thrust of the project is to continue joint work with John Shareshian on the interplay between symmetric function theory, enumerative combinatorics, and algebraic geometry. This project arose from the study of a class of symmetric functions that appears in various contexts such as in the work of Carlitz, Scoville and Vaughan on enumeration of Smirnov words, in the work of Procesi and Stanley on the representation of the symmetric group on the cohomology of the toric variety associated with the type A root system, and in the work of the investigator and Shareshian on a q-analog of the Eulerian polynomials. In an effort to understand the basis for the relationship between these structures, the investigator and Shareshian have proposed a far-reaching generalization of this relationship, which this project will explore. The conjectured generalization involves a quasisymmetric refinement of Stanley's chromatic symmetric functions, Tymoczko's representation of the symmetric group on the cohomology of the regular semisimple Hessenberg variety of type A, and a q-analog of a generalization (due to De Mari, Procesi and Shayman) of the Eulerian polynomials. The conjectured relationship would provide an algebro-geometric approach to attacking Stanley and Stembridge's long standing $e$-positivity conjecture for chromatic symmetric functions and a conjecture of the investigator and Shareshian on unimodality of their q-analog of the De Mari-Procesi-Shayman generalized Eulerian polynomials. The project will also continue the work of the investigator on the interplay between poset topology and enumerative combinatorics.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 increasingly important role in these fields.