Research in Algebraic Combinatorics

我的研究计划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。我的目标之一是扩大和汇集最近的重要发展前沿的四个领域，即：代数（对角模块的多项式），特殊功能（运营商的对称函数），组合学（矩形戴克路径和停车功能），和结理论（绞代数的（m，n）-环面结）。几年来，我一直处于前三个领域研究的最前沿，我的项目位于关于它们相互作用的主要开放问题的中心。此外，在这些领域中的每一个领域都有令人兴奋的最新发展，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，我目前正在这些领域取得重大进展。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，最近我明确地概述了这种奋进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要方面提出的： 1. Macdonald多项式的组合数学，相关的算子和相关的模，与矩形Catalan组合数学的联系，以及它们与椭圆Hall代数的联系， 2.在几组变量中的组合和表示理论类似物， 3.对称函数体积的性质和福克斯猜想。 第二种方法利用了我几年前提出的想法，关于将问题扩展到多对角版本，这些问题在双对角情况下是如此富有成效。关于这一点，值得强调的是，自20世纪90年代中期以来，在顶级期刊上出现了100多篇与这种情况（k=2）有关的重要论文。扩展到多对角情形（k>2）必然会使这一研究影响倍增，并给出更多的根本原因，说明为什么所有这些都孕育着重要的新知识。 我还提出了原创的新技术来构建理论中涉及的组合结构和对称函数对象的代数对应物（多项式模块）。这一直是这个研究领域长期以来缺失的重要部分。我的新方法一定会为这一背景下的一个基本主题提供一个令人满意的解释：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。 这最后一个方面把我带到了我的计划的最后一部分，关于许多新的方法来理解一个猜想的福克斯，可以追溯到近70年。特别是，我提出了一个新的和原始的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.