Combinatorics and Representation Theory

本提案旨在探讨组合学与表示理论之间相互作用的几个方面。首先，PI将研究由Reiner， Stanton和White引入的计数循环筛分现象，因为它适用于Fomin和Zelevinsky的簇状配合物的组合作用，以及由晶体理论产生的对表的作用。PI将使用表示理论中的对象，如Kazhdan-Lusztig基和聚类代数来研究这些组合问题。其次，PI将研究超平面排列和与对角协不变模相关的某些环的希尔伯特级数之间的联系。有限集合中对象的枚举是组合学的基本问题，在纯数学之外有许多科学应用。对称群是由古希腊人提出的，是表征理论研究的主要动机。因此，枚举对称性的研究在组合表示理论和代数组合学中具有极其重要的意义。这个项目使用表征理论来间接计算有限集合的元素，从而对它们的枚举产生更深入、更优雅的理解。

This proposal aims to investigate several aspects of the interplay between combinatorics and representation theory. First, the PI will investigate the enumerative cyclic sieving phenomenon introduced by Reiner, Stanton, and White as it applies to combinatorial actions on the cluster complexes of Fomin and Zelevinsky as well as actions on tableaux arising from the theory of crystals. The PI will use objects from representation theory such as Kazhdan-Lusztig bases and cluster algebras to investigate these combinatorial problems. Second, the PI will investigate connections between hyperplane arrangements and the Hilbert series of certain rings related to the diagonal coinvariant module.The enumeration of objects in a finite set is the fundamental problem of combinatorics and has many scientific applications outside of pure mathematics. Symmetry groups were introduced by the ancient Greeks and are the primary motivation for the study of representation theory. So, the study of enumeration up to symmetry is of paramount importance in combinatorial representation theory and algebraic combinatorics. This project uses representation theory to indirectly count the elements of finite sets, yielding a deeper and more elegant understanding of their enumeration.