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、表示论、拓扑学、数学物理和统计力学等许多学科有许多联系。我更感兴趣的组合和几何方面的研究考克斯特（镜面反射）集团及其相关结构。考克斯特群出现在数学的许多领域，例如，作为正则多面体的对称群，作为半单李代数和卡茨-穆迪代数的外尔群，以及作为几何中的三角群（欧几里得和双曲）。这些基团的性质通常是理解相关结构的关键。 众所周知，根系是考克斯特群理论的基础。虽然有限和仿射根系统已经得到了很多关注，但对于一般的无限根系统几乎一无所知。我最近发现，与J。P.~ Labbé（柏林），V.~ Ripoll（UQAM），一种令人兴奋的新方法，包括研究根的极限点，在几个方向上开辟了一个大型研究项目，每个方向都值得独立研究。这些方向去从应用到卡茨-穆迪代数的类似物的广义associahedra在无限的情况下，包括推广弱秩序的根系。而广义结合面体是研究团代数的基本几何对象，其分支扩展到物理学、热力学和统计学等领域。我计划利用我的研究成果。另一条线的研究，我将追求与考克斯特群的研究下降代数。 下降代数是有限Coxeter群表示论的丰富版本中的关键成分。我计划通过利用我过去在对称群和超八面体群上的工作，以及基于“D型霍普夫代数”的新思想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".