Combinatorics, Representations, and Catalan Theory

本课题研究数列组合学与代数组合学交界的问题。组合问题出现在数学的许多领域，组合学的应用包括优化、计算机科学和统计物理。本项目研究的枚举问题与停放函数（最初是在计算机科学的哈希函数研究中出现的）和循环筛分现象（枚举组合学中的一个概念，将某些多项式计算解释为不动点的计数）有关。该研究旨在证明枚举组合学的结果，并从更深的代数结构方面理解这些结果。组合学和代数之间的这种相互作用有望在这两个领域产生新的结果。研究的列举方面非常适合以研究生和本科生研究项目的形式产生更广泛的影响。本课题研究代数组合学中的问题。其中第一个是循环筛分现象，因为它适用于提升算子的k -理论模拟对矩形标准杨氏表的k -泛化的作用。这个想法是证明（枚举）循环筛分现象的新实例相关的行动使用表示理论。第二个问题是将对称群上的停车函数推广到反射群w上的更广泛的“停车空间”类。我们研究了关于这些对象的一系列猜想，这些猜想将产生coxet - catalan理论中各种事实的一致证明，这些事实目前只能以个案的方式来理解。第三个课题研究理性加泰罗尼亚组合，这是经典加泰罗尼亚组合的推广，是由理性Cherednik代数的研究推动的。我们建议将经典设定的丰富枚举领域的各种结果扩展到理性情况，并研究理性情况的一个真正的新特征，我们称之为“理性对偶性”。

This project studies problems at the interface of enumerative and algebraic combinatorics. Combinatorial questions arise in many areas of mathematics, and combinatorics has applications that include optimization, computer science, and statistical physics. The enumerative problems under study in this project are related to parking functions (which originally arose in the study of hash functions in computer science) and the cyclic sieving phenomenon (a concept in enumerative combinatorics that interprets certain polynomial evaluations as counts of fixed points). The research aims to both prove results in enumerative combinatorics and understand these results in terms of deeper algebraic structures. This interaction between combinatorics and algebra promises to yield new results in both fields. The enumerative side of the research is well-suited to broader impacts in the form of graduate and undergraduate research projects.This project studies problems in algebraic combinatorics. The first of these is the cyclic sieving phenomenon as it applies to the action of a K-theoretic analog of the promotion operator on a K-generalization of rectangular standard Young tableaux. The idea is to prove new instances of the (enumerative) cyclic sieving phenomenon related to this action using representation theory. The second problem concerns a generalization of parking functions attached to the symmetric group to a wider class of "parking spaces" attached to a reflection group W. We study a family of conjectures regarding these objects which would yield uniform proofs of various facts in Coxeter-Catalan theory which are at present only understood in a case-by-case fashion. The third project studies rational Catalan combinatorics, which is a generalization of classical Catalan combinatorics motivated by the study of rational Cherednik algebras. We propose to both extend various results from the rich enumerative domain of the classical setting to the rational case and study a genuinely new feature of the rational case that we call "rational duality."