Research in Algebraic Combinatorics

代数组合学研究

• 批准号: RGPIN-2017-05104
• 负责人: Bergeron, François
• 金额: $ 2.7万
• 依托单位: Université du Québec à Montréal
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=645241
• 关键词: Research; Algebraic; Combinatorics

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.

我的研究项目处于代数和组合学的前沿，特别强调有限群的表示论及其不变量，以及与代数几何、拓扑、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前沿的最新重要发展，即：代数（多项式的对角模）、特殊函数（对称函数的运算符）、组合数学（矩形戴克路径和停车函数）以及结理论（(m,n)-环面结的绞纱代数）。多年来，我一直处于前三个领域研究的前沿，我的项目位于有关它们相互作用的主要开放问题的中心。此外，这些领域最近都取得了令人兴奋的发展，同时也出现了大量深刻的问题。我与我的学生、博士后和合作者一起，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这种互动中需要解决的许多核心问题，最近我已经明确概述了这项努力现在应该走向哪些主要方向。更明确地说，我的建议围绕以下四个主轴进行阐述：*** 1. 麦克唐纳多项式的组合、相关运算符和相关模块，以及与矩形加泰罗尼亚组合的链接及其与椭圆霍尔代数的联系，*** 2 . 多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，以及福克斯猜想。*** 第二个利用了我几年前提出的关于将问题扩展到多对角版本的想法，这些问题在双对角案例中非常富有成效。关于这一点，可能值得强调的是，自 20 世纪 90 年代中期以来，与此案例 (k=2) 有关的顶级期刊中肯定有一百多篇重要论文发表。扩展到多对角线情况 (k>2) 必然会增加这项研究的影响，并给出更根本的原因，解释为什么所有这些都如此孕育着重要的新知识。*** 我还提出了原创的新技术为理论中涉及的组合结构和对称函数对象构造代数对应项（多项式的模）。这是该研究领域长期缺失的重要部分。我的新方法一定会为这种背景下的基本主题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； 并解释为什么这种积极现象如此占主导地位。*** 这最后一个方面让我进入了我的计划的最后一部分，关于理解福克斯猜想的许多新方法，该猜想可以追溯到近 70 年前。我特别提出了一种新的、原创的 q 模拟。