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群论，我最喜欢的主题是考克斯特群，反思群及其相关结构。我喜欢的问题发生在代数和几何组合学的界面上。 考克斯特群在数学的几个领域中扮演着重要的角色：它们作为Weyl群出现在李理论、簇代数或代数几何中;它们是作用于几何中常曲率空间的离散反射群，它们是几何群论中定义建筑物的基础;它们也自然地出现在理论物理、化学和生物信息学中。这些组的属性通常是深入理解这些区域的主要相关结构的关键。 虽然有限Coxeter群或其所扮演的角色有大量丰富的文献，但相比之下，无限Coxeter群的情况由于缺乏明确的组合工具而在很大程度上尚未探索。我的研究计划是关于感受我们对无限Coxeter群的精细结构的理解中的差距。 在过去的6年里，我和我的合作者、博士后和学生一起，一直站在无限Coxeter群组合学新方法的最前沿。我们的工作提供了工具，澄清和揭示这些群体的组合，几何和拓扑方面之间的深刻联系。我目前的研究计划是这项工作的直接延续，围绕以下三个轴进行阐述： 1)无限根系与极限根; 2)弱序、Bruhat序与双闭根集; 3)在考克斯特群体中的Garside阴影。 考克斯特群研究的核心是它们的抽象定义和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刻联系。 在我的研究计划的不同组成部分之间的一个共同的技术线索是弱序，它有一个自然的几何解释，这是重要的考克斯特群整除是整数。 第一个主题是关于加强桥梁，我已经作出贡献，以建设之间的代数组合学和几何群论的观点对考克斯特群 第二个主题是关于两个美丽的插图，旨在加深我们对Bruhat阶，Hecke代数和Kazhdan-Lusztig多项式的理解。我的动机在这方面是设计一个集群/加泰罗尼亚组合理论的无限情况下。 最后一个方向是探讨Garside阴影的概念，我和我的合作者在研究Artin-Tits Braid群中的词和共轭问题时已经证明了这一点。作为结果，我们的目标是简化描述的Coxeter群的自动结构，其目的是打开一个新的视角研究仍然开放的问题的Coxeter群的双自动性。