Combinatorics, Representations, and Catalan Theory

本计画研究计数组合学与代数组合学之介面问题。 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。 在这个项目中研究的枚举问题与停车函数（最初出现在计算机科学中的哈希函数研究中）和循环筛选现象（枚举组合学中的一个概念，将某些多项式求值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，并从更深层次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。 该研究的枚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。 其中第一个是循环筛选现象，因为它适用于行动的K-理论模拟的推广运营商的K-推广的矩形标准杨tableaux。 我们的想法是证明新的例子（枚举）循环筛选现象与此行动使用表示论。 第二个问题涉及到一个广义的停车场功能连接到对称群到一个更广泛的类的“停车位”连接到一个反射组W。 我们研究一个家庭的apturtures关于这些对象，将产生统一的证明，各种事实，在考克斯特-加泰罗尼亚理论，目前只了解在一个个案的方式。 第三个项目研究理性加泰罗尼亚组合，这是一个推广的经典加泰罗尼亚组合的动机研究理性切雷德尼克代数。 我们建议将各种结果从经典环境的丰富枚举域扩展到理性情况，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."