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这种离散结构的例子。组合方法在这些领域中发挥着越来越重要的作用。提出的研究包括三个相关的项目，涉及经典组合对象的推广和变化，如色多项式、欧拉多项式、分格和自由李代数。第一个项目涉及斯坦利的色对称函数的细化，该函数是在John Shareshian和PI的工作中引入的。其改进，即色拟对称函数，推广了欧拉多项式和色多项式。在这项工作中，提出了一种代数几何方法来解决长期存在的Stanley-Stembridge e正猜想，该猜想涉及到色拟对称函数与De Mari， Procesi和Shayman的Hessenberg变异之间的联系。这种方法在研究对称函数的组合学家和研究海森伯格变分的代数几何学家之间引发了重要的互动。进一步研究了色拟对称函数及其与Hessenberg变分的关系。第二个项目涉及到PI和她以前的学生Rafael Gonzalez D'Leon引入的一个新的多项式图不变量和一个更一般的对称函数图不变量。这些图不变量推广了欧拉多项式的变体，如经典的Narayana多项式和Haiman的停车函数对称函数。本课题的重点是解决这些图不变量的单模猜想、e-正猜想和其他具体猜想。第三个项目产生于理论物理学。它处理的是李代数的n元泛化，也就是菲利波夫代数。自由Filippov代数的多线性分量上对称群表示的研究是在Friedmann、Hanlon、Stanley和PI的工作中开始的。虽然这项工作已经产生了一些重大成果，但仍有许多工作要做。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The research supported by this grant is in algebraic combinatorics, 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 such discrete configurations. Combinatorial methods are playing an increasing role in these fields. The proposed research is comprised of three related projects involving generalizations and variations of classical combinatorial objects such as chromatic polynomials, Eulerian polynomials, partition lattices, and free Lie algebras. The first project deals with a refinement of Stanley's chromatic symmetric functions that was introduced in work of John Shareshian and the PI. The refinement, the chromatic quasisymmetric functions, generalizes Eulerian polynomials as well as chromatic polynomials. An algebro-geometric approach to settling the longstanding Stanley-Stembridge e-positivity conjecture, involving a connection between the chromatic quasisymmetric functions and the Hessenberg varieties of De Mari, Procesi, and Shayman, was presented in this work. This approach initiated significant interaction between combinatorialists working with symmetric functions and algebraic geometers working on Hessenberg varieties. Further study of the chromatic quasisymmetric functions, as well as its connection with Hessenberg varieties is proposed. The second project involves a new polynomial graph invariant and a more general symmetric function graph invariant introduced by the PI and her former student Rafael Gonzalez D'Leon. These graph invariants generalize variations of Eulerian polynomials such as the classical Narayana polynomials and Haiman's parking function symmetric functions. This project is focused on settling unimodality conjectures, e-positivity conjectures and other concrete conjectures for these graph invariants. The third project arose in theoretical physics. It deals with an n-ary generalization of a Lie algebra known as a Filippov algebra. The study of the representation of the symmetric group on the multilinear component of the free Filippov algebra was initiated in work of Friedmann, Hanlon, Stanley, and the PI. Although this work has already produced some significant results, there is much that remains to be done.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.