Coxeter groups and related structures

我的主要研究领域是代数组合学。它是一个高度活跃的数学领域，与代数几何、凸几何、表示理论、拓扑学、数学物理和统计力学等许多学科都有联系。我对Coxeter（镜像反射）群及其相关结构的组合和几何方面的研究更感兴趣。Coxeter群出现在许多数学领域，例如，作为正多面体的对称群，半简单李代数和Kac-Moody代数的Weyl群，以及几何（欧几里得和双曲）中的三角形群。这些基团的性质往往是理解相关结构的关键。根系统是考克斯特群理论的基础，这是公认的。虽然有限根和仿射根系统已经得到了很多关注，但对于一般无限根系统几乎一无所知。我最近发现了j。p。里波尔（UQAM），一种令人兴奋的新方法，它包括对根的极限点的研究，在几个方向上开辟了一个大的研究计划，每个方向都值得独立研究。这些方向从对Kac-Moody代数的应用到无限情况下广义结合体的类似，并包括根上弱序的推广。而广义关联面是聚类代数研究中的基本几何对象，其分支延伸到物理学、热力学和统计学。我打算利用我的研究成果。我将进行的另一项研究将考克斯特群与下降代数的研究联系起来。下降代数是有限Coxeter群表示理论的一个丰富版本的关键成分。我计划利用我过去在对称群和超八面体群上的工作，以及基于“D型Hopf代数”的新想法，来揭示这种丰富的结构。

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".