Catalan Structures for Weyl and Coxeter Groups

该建议是研究四个重要的组合结构相关的加泰罗尼亚号码（associahedra，非交叉分区，非嵌套分区，Tamari格），其中一些最近已被推广到所有有限的Weyl群的A型。 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。该提案的重点是对对称性研究至关重要的对象，以及它如何在“自然界”中发生，这在数学中被称为“表示论”。 这个建议有望加深我们对这种对称性的理解，解释更多的特征，以及我们如何利用它们。

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.