Combinatorics, Representations, and Catalan Theory

组合学、表示法和加泰罗尼亚理论

• 批准号: 1500838
• 负责人: Brendon Rhoades
• 金额: $ 18万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-07-01 至 2021-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1500838&HistoricalAwards=false
• 关键词: Combinatorics; Representations; Catalan; Theory

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."

该项目研究了枚举和代数组合学的界面上的问题。 组合问题在数学的许多领域都会出现，并且组合技术具有包括优化，计算机科学和统计物理学的应用。 该项目中研究的列举问题与停车功能（最初是在计算机科学中的哈希功能研究中引起的）和循环筛分现象（在枚举组合中的概念，将某些多项式评估解释为固定点计数）。 该研究旨在证明列举组合学的结果，并根据更深的代数结构来理解这些结果。 组合物与代数之间的这种相互作用有望在这两个领域都产生新的结果。 这项研究的列举方面非常适合以研究生和本科研究项目的形式更广泛的影响。该项目研究问题在代数组合学中。 其中首先是循环筛分现象，它适用于促进操作员对矩形标准幼型tableaux的K将其化为k的keneralization的作用。 这个想法是证明使用表示理论与此行动相关的（列举）环状筛分现象的新实例。 第二个问题涉及对称组附带的停车功能的概括，以与反射群体相关的更广泛的“停车位”。我们研究了有关这些物体的猜想家族，这些家族将产生Coxeter-的各种事实的统一证明 - 目前，加泰罗尼亚理论仅以逐案方式理解。 第三个项目研究了有理加泰罗尼亚联合术，这是对理性Cherednik代数的促进的经典加泰罗尼亚组合制剂的概括。 我们建议将各种结果从经典环境的丰富枚举领域扩展到理性案例，并研究我们称之为“理性二元性”的理性案例的真正新特征。