Algebraic and geometric combinatorics of Coxeter groups

Coxeter 群的代数和几何组合

• 批准号: RGPIN-2018-04615
• 负责人: Hohlweg, Christophe
• 金额: $ 1.68万
• 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: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=712852
• 关键词: Algebraic; geometric; combinatorics; Coxeter; groups

My primary area of research is group theory, and my favorite themes are Coxeter groups, reflections groups and their related structures. My preferred questions take place at the interface of algebraic and geometric combinatorics. Coxeter groups play a fundamental role in several areas of mathematics: they occur as Weyl groups in Lie theory, for Cluster algebras or in algebraic geometry; they are the discrete reflection groups acting on spaces of constant curvature in geometry and they are fundamental to define buildings in geometric group theory; they also occur naturally in Theoretical Physics, Chemistry and Bioinformatics. Properties of these groups are often key to a deep understanding of the main relevant structures for these areas. While there is a vast and rich literature on finite Coxeter groups, or on the role they play, the case of infinite Coxeter groups is, in comparison, still largely unexplored due to lack of explicit combinatorial tools to do so. My research program is about feeling the gap in our understanding of the fine structure of infinite Coxeter groups. Together with my collaborators, postdocs and students, I have been at the forefront of a new approach to the combinatorics of infinite Coxeter groups for the past 6 years. Our work provides tools that clarify and reveal profound ties between combinatorial, geometrical and topological aspects of these groups. My current research program, which is a direct continuation of this work, is articulated around the following three axes: 1) Infinite root systems and limit roots; 2) Weak order, Bruhat order and biclosed sets of roots; 3) Garside shadows in Coxeter groups. At the core of the study of Coxeter groups is a deep connection between their abstract definition and their geometric realizations as reflection groups acting on some geometric spaces. A common technical thread between different components of my research program is the weak order, which has a natural geometric interpretation, and which is as important for Coxeter groups as divisibility is for the integers. The first topic is about strengthening the bridge that I have contributed to construct between the algebraic combinatorics and the geometric group theory points of view on Coxeter groups The second topic is concerned with two beautiful conjectures designed to deepen our understanding of Bruhat order, Hecke algebras and Kazhdan-Lusztig polynomials. My motivation in this context is to design a Cluster/Catalan combinatorics theory for the infinite case. The last direction is concerned with exploring the notion of Garside shadows that my collaborators and I have made evident in relation to the study of the word and conjugacy problems in Artin-Tits Braid' groups. As consequences, we aim to simplify the description of the automatic structure of Coxeter groups, which aim to open a new perspective on the study of the still open problem of bi-automaticity for Coxeter groups.

我的主要研究领域是小组理论，我最喜欢的主题是Coxeter组，反射组及其相关结构。我首选的问题发生在代数和几何组合学的界面。 Coxeter群在数学的几个领域中起着基本作用：它们以谎言理论，群集代数或代数几何形状作为Weyl群体出现；它们是作用于几何恒定曲率空间的离散反射组，它们对于在几何群体理论中定义建筑物至关重要。它们也自然出现在理论物理，化学和生物信息学中。这些群体的特性通常是对这些领域主要相关结构的深入了解的关键。 虽然有关于有限的高克西特群体或它们所扮演的角色的丰富文献，但相比之下，无限的高级汽车组的情况仍然在很大程度上没有探索，这是由于缺乏明确的组合工具来做到这一点。我的研究计划是关于我们对无限Coxeter群体良好结构的理解的差距。 在过去的6年中，我与我的合作者，博士后和学生一起一直处于无限Coxeter组组合的最前沿。我们的工作提供了这些群体的组合，几何和拓扑方面之间澄清和揭示紧密联系的工具。我目前的研究计划是这项工作的直接延续，围绕以下三个轴阐明： 1）无限根系并限制根部； 2）较弱的顺序，布鲁哈特秩序和双根集； 3）Coxeter组中的Garside阴影。 Coxeter组研究的核心是它们的抽象定义与它们作为作用于某些几何空间的反射群的几何实现之间的深厚联系。 我的研究计划不同组成部分之间的一个常见技术线程是弱顺序，它具有自然的几何解释，对于Coxeter群体而言，这与整数一样重要。 第一个主题是加强我为代数组合和几何组理论在Coxeter群体上的观点之间构建的桥梁 第二个主题与两个美丽的猜想有关，旨在加深我们对Bruhat Order，Hecke代数和Kazhdan-Lusztig多项式的理解。在这种情况下，我的动机是为无限案例设计群集/加泰罗尼亚组合理论。 最后一个方向是探索我和我的合作者在Artin-tits Braid'群中的研究和共轭问题的研究方面显而易见的Garside阴影概念。结果，我们旨在简化Coxeter群体的自动结构的描述，该结构旨在对Coxeter群体的Bi-Autamationity仍然开放的新观点。