Research in Algebraic Combinatorics

我的研究项目位于代数和组合学的前沿，特别强调有限群及其不变量的表示理论，以及与代数几何、拓扑学、数学物理和统计力学的联系。我的目标之一是扩展和汇集四个领域的最新重要发展，即：代数(多项式的对角模)、特殊函数(对称函数上的算子)、组合学(矩形Dyck路和停车函数)和纽结理论((m，n)环面纽结的skein代数)。几年来，我一直站在这些领域中前三个领域的研究前沿，我的项目位于关于它们相互作用的主要公开问题的中心。此外，在这些领域中的每一个领域都有令人振奋的最新事态发展，并出现了大量深刻的问题。与我的学生、博士后和合作者一起，我目前在这些领域都取得了重大进展。事实上，我们的工作已经澄清了在这种互动中需要解决的许多核心问题，最近我明确提出了这项努力现在应该朝着哪些主要方向发展。更明确地说，我的建议是围绕以下四个主轴提出的：*1.麦克唐纳多项式、相关算子和相关模的组合学，以及与矩形加泰罗尼亚组合学的联系和它们与椭圆霍尔代数的联系，*2.几组变量中的组合和表示理论类比，*3.对称函数的多解性的性质，以及Foulkes猜想。*第二个利用了我几年前提出的关于将双对角线情况下卓有成效的问题扩展到多对角线版本的想法。关于这一点，可能值得强调的是，自20世纪90年代中期以来，肯定有100多篇重要论文发表在与此案(k=2)有关的顶级期刊上。对多对角线情形(k&gt；2)的扩展必然会增加这一研究的影响，并给出更根本的原因，为什么所有这些都充满了重要的新知识。*我还提出了原创性的新技术来构造组合结构和对称函数对象的代数副本(多项式的模)。这一直是这一研究领域的一个重要缺失部分。我的新方法一定会为这个背景下的一个基本主旨提供一个令人满意的原创解释：所涉及的对称函数的舒尔正性；并解释为什么这种正性现象如此突出。*这最后一个方面将我带到我的程序的最后部分，关于许多新的方法来理解一个可以追溯到近70年前的福克斯猜想。特别是，我提出了一种新的、独创的Q模拟。

My program of research sits at the frontier of algebra and combinatorics, with specific emphasis on representation theory of finite groups and their invariants, and connections with algebraic geometry, topology, mathematical physics and statistical mechanics. One of my aims is to expand and bring together recent important developments at the frontier of four areas, namely: algebra (diagonal modules of polynomials), special functions (operators on symmetric functions), combinatorics (rectangular Dyck paths and parking functions), and knot theory (skein algebra of (m,n)-torus knot). For several years, I have been at the forefront of research in the first three of these areas, and my project sits at the very center of main open questions regarding their interactions. Moreover, there are exciting recent developments in each of these areas, and a large number of profound problems are arising. Together with my students, postdocs and collaborators, I am currently making significant progress in each of these areas. Indeed, our work has clarified many of the central questions that need to be solved in this interaction, and more recently I have explicitly outlined which main directions this endeavor should now go toward. More explicitly, my proposal is articulated around the four following main axes:*** 1. Combinatorics of Macdonald polynomials, related operators, and associated modules, and links with rectangular Catalan combinatorics and their connections to the elliptic Hall algebra,*** 2. Combinatorial and representation theoretic analogs in several sets of variables,*** 3. Properties of plethysms of symmetric functions, and the Foulkes conjecture.***The second of these exploits ideas that I proposed a few years back concerning the expansion to multidiagonal versions of the questions that have been so fruitful in the bidiagonal case. About this, it may be worth underlining that certainly more than a hundred significant papers have appeared in top journals in relation to this case (k=2) since the mid 1990s. An expansion to the multidiagonal case (k>2) is bound to multiply this research impact, as well as give more fundamental reasons why all of this is so pregnant with significant new knowledge.*** I have also come up with original new techniques to construct the algebraic counterparts (modules of polynomials) for the combinatorial constructions and symmetric function objects involved in the theory. This has been a long-standing important missing part in this research area. My new approach is bound to furnish an original satisfying explanation for a fundamental leitmotif in this context: the Schur positivity of the symmetric functions involved; and explain why this positivity phenomenon is so predominant.*** This last aspect brings me to the last portion of my program regarding many new ways of understanding a conjecture of Foulkes that dates back almost 70 years. In particular, I propose a new and original q-analog.