Catalan Structures for Weyl and Coxeter Groups

研究与Catalan数有关的四种重要的组合结构(结面体、非交叉划分、非嵌套划分、Tamari格)，其中一些最近已从A型推广到所有有限Weyl群。其目标是通过双射将它们联系在一起，并以统一的方式理解它们的枚举、代数和拓扑性质。该提案的重点是研究对称性的核心对象，以及它如何在自然界中发生，这是一门在数学上被称为“表象理论”的学科。这项提案有望加深我们对这种对称性的理解，解释它发生时伴随的更多特征，以及我们可能如何利用它们。

The proposal is to study four important combinatorial structures related to Catalan numbers (associahedra, non-crossing partitions, non-nesting partitions, Tamari lattices), some of which have recently been generalized from type A to all finite Weyl groups. The goal is to tie them together via bijections and understand their enumerative, algebraic, and topological properties in a unified way.The focus of the proposal is on objects central to the study of symmetry, and how it can occur "in nature", a subject known in mathematics as "representation theory". The proposal will hopefully deepen our understanding of such symmetry, by explaining more of the features that accompany it whenever it occurs, and possibly how we can exploit them.