Combinatorics, Representations, and Catalan Theory

这个项目研究了枚举和代数组合数学的交界处的问题。组合问题出现在数学的许多领域，组合数学的应用包括最优化、计算机科学和统计物理。这个项目中研究的计数问题与停车函数(最初出现在计算机科学中对哈希函数的研究)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计数)有关。这项研究的目的是既证明列举组合学中的结果，又用更深层次的代数结构来理解这些结果。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。该项目研究代数组合学中的问题。第一个是循环筛选现象，因为它适用于提升算子的K-理论模拟在矩形标准Young表的K-推广上的作用。其想法是使用表示理论来证明与该动作相关的(枚举式)循环筛选现象的新实例。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“停车位”类。我们研究了一族关于这些物体的猜想，这些猜想将给出Coxeter-Catalan理论中的各种事实的统一证明，而这些事实目前只能以个案的方式来理解。第三个项目研究有理加泰罗尼亚组合学，这是经典加泰罗尼亚组合学的推广，其动机是有理Cherednik代数的研究。我们建议将各种结果从经典背景下丰富的枚举域扩展到理性情形，并研究理性情形的一个真正的新特征，我们称之为“理性对偶”。

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."