Coxeter groups and related structures

考克塞特群及相关结构

• 批准号: 355458-2013
• 负责人: Hohlweg, Christophe
• 金额: $ 1.38万
• 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: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=572578
• 关键词: Coxeter; groups; related; structures

My primary area of research is algebraic combinatorics. It is a highly active area of mathematics, with many connections to algebraic geometry, convex geometry, representation theory, topology, mathematical physics and statistical mechanics, among many others. I am more precisely interested on the combinatorial and geometrical aspects of the study of Coxeter (mirror reflection) groups and their related structures. Coxeter groups appear in very many domains of mathematics, for instance, as symmetry groups of regular polytopes, as Weyl groups of semi-simple Lie algebras and Kac-Moody algebras, and as triangle groups in geometry (Euclidean and hyperbolic). Properties of these groups are often key to the understanding of related structures. It is well-established that root systems are fundamental in the theory of Coxeter groups. While finite and affine root systems have been given a lot of attention, almost nothing is known for general infinite root systems. I have uncovered recently, with J.-P.~Labbé (Berlin), V.~Ripoll (UQAM), an exciting new approach that consists in the study of the limit points of roots, opening up a large program of research in several directions, each worthy of independant study. These directions go from applications to Kac-Moody algebras to analogs of generalized associahedra in the infinite case and include generalizations of weak order on root systems. whereas generalized associahedra are fundamental geometric objects in the study of cluster algebras, whose ramifications extend to physics, thermodynamics and statistics. I plan to exploit my research. Another line of research I will pursue relate Coxeter groups with the study of Descent algebras. Descent algebras are key ingredients in an enriched version of the representation theory of finite Coxeter groups. I plan to uncover this enriched structure by exploiting my past work on symmetric groups and hyperoctahedral groups, together with a new idea based on a "type D Hopf algebra".

我的主要研究领域是代数组合学。它是数学高度活跃的领域，与代数几何形状，凸几何，表示理论，拓扑，数学物理学和统计力学等有许多联系。我对Coxeter研究（镜像）组及其相关结构的研究的组合和几何方面更加精确感兴趣。 Coxeter组出现在数学的许多领域中，例如，作为常规多型的对称组，如半简单的lie代数和KAC-MOODY代数，以及几何形状（欧几里得和双胞胎）中的三角形基团。这些组的特性通常是对相关结构的理解的关键。 良好建立的根系在Coxeter群体的理论中是基础。尽管有限和仿射根系受到了很多关注，但几乎没有任何一般无限根系所知道的。我最近发现了J.-P。〜labbé（柏林），V。这些方向从无限情况下从应用到KAC-MOODY代数到广义联想的类似物，包括对根系的弱点的概括。而广义的联想是群集代数研究中的基本几何对象，其影响扩展到物理，热力学和统计数据。我计划利用我的研究。我将通过对下降代数的研究进行相关的Coxeter组。下降代数是有限的Coxeter组的代表理论的重要版本中的关键因素。我计划通过利用我过去的对称群体和高核心群的工作以及基于“ D型Hopf代数”的新想法来揭示这种丰富的结构。