Coxeter-Catalan Combinatorics

加泰罗尼亚组合学是代数和几何组合学中一个非常活跃的研究领域，与不变量理论、代数几何和表示论等其他领域有着多重联系。一个关键因素是统一地描述在各个领域中可以发现的现象，从而显示出先验的无关概念之间的关系。这些研究领域的共同点是，它们可以根据卡坦杀戮类型进行分类，对于这些类型，通过推广著名的加泰罗尼亚数字来计算的自然对象是基本感兴趣的。在这个项目中，研究了以下几个方面之间的联系：(1)根系统，(2)反射群，(3)簇代数，(4)子词复合体。该项目的目标是进一步确定它们的组合性质和联系。这些工具是来自交换和非交换代数的代数技巧，来自根系统、超平面排列和单纯复形的几何技巧，来自特征标理论的表示理论技巧，来自对称函数论的组合技巧，以及来自枚举和双射组合数学的组合技巧。

Catalan combinatorics is a highly active area of research within algebraic and geometric combinatorics, with multiple connections to other fields like invariant theory, algebraic geometry, and representation theory. A key ingredient is to uniformly describe phenomena which can be found in the various fields, and thereby exhibiting relations between a priori unrelated concepts. Those fields of research have in common that they can be classified according to Cartan-Killing types, and for which natural objects counted by generalizations of the famous Catalan numbers are of fundamental interest. In this project, connections between the following avenues of research are studied: (1) root systems, (2) reflection groups, (3) cluster algebras, (4) subword complexes. The objective of the project is to further establish their combinatorial properties and connections. The tools are algebraic techniques from commutative and noncommutative algebra, geometric techniques from root systems, hyperplane arrangements and simplicial complexes, representation theoretical techniques from character theory, and combinatorial techniques from symmetric function theory, and from enumerative and bijective combinatorics.