Combinatorics and Braid Varieties

组合学和编织品种

• 批准号: 2246877
• 负责人: Nathan Williams
• 金额: $ 21万
• 依托单位: University of Texas at Dallas
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2246877&HistoricalAwards=false
• 关键词: Combinatorics; Braid; Varieties

Algebraic combinatorics is a branch of mathematics that studies algebraic structures using combinatorial methods, and combinatorial structures using algebraic methods. Such structures and methods are often fundamental to many different scientific disciplines and show up in many different contexts---algebraic combinatorics therefore has diverse applications to subjects like cryptography, protein folding, high-energy physics, and quantum computing. This project will use algebraic objects and methods to produce new combinatorial results, leveraging braid varieties--a sort of algebraic space associated to a knot--as a unifying tool. Funds will additionally support training graduate students and outreach efforts, including work on an interactive online discrete mathematics textbook.In more detail, this proposal suggests a framework for producing combinatorial results using braid varieties over finite fields, Hecke algebra traces, rational Cherednik algebras, and a new relationship with noncrossing combinatorics. The framework has already proven successful in producing substantial new results: the PI's recent joint work with Galashin, Lam, and Trinh resolved two decades-long open problems in Coxeter-Catalan combinatorics, simultaneously producing the first definition of rational noncrossing Coxeter-Catalan objects, while also giving the first uniform enumeration of noncrossing objects. Connections to Macdonald theory--diagonal harmonics and q,t-combinatorics--are also expected when working over the complex numbers. At different levels of generality, different techniques become available. For finite Coxeter groups, it is possible to compute everything in a case-by-case manner using an explicit decomposition of the Hecke algebra, and there are many interesting combinatorial and representation-theoretic problems open for immediate attack. Special classes of elements in finite type have favorable representation-theoretic properties that allow for uniform approaches. For affine Weyl groups, the main tool is a trace formula for translations, due to Opdam. For example, the proposed framework recovers some Tessler matrix identities due to Haglund in this setting. For general Kac-Moody Weyl groups, we are reduced to general recursive and cluster-theoretic methods. These methods also apply in both the finite and affine cases, but will require software implementation before further exploration is possible.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

代数组合学是数学的一个分支，它使用组合方法研究代数结构，以及使用代数方法研究组合结构。 这些结构和方法通常是许多不同科学学科的基础，并出现在许多不同的环境中——因此，代数组合学在密码学、蛋白质折叠、高能物理和量子计算等学科中具有多种应用。 该项目将使用代数对象和方法来产生新的组合结果，利用辫子变种（一种与结相关的代数空间）作为统一工具。 资金还将支持研究生培训和推广工作，包括交互式在线离散数学教科书的工作。更详细地说，该提案提出了一个使用有限域上的辫子簇、赫克代数迹、有理切雷德尼克代数以及与非交叉组合学的新关系来产生组合结果的框架。 事实证明，该框架已经成功地产生了实质性的新成果：PI 最近与 Galashin、Lam 和 Trinh 的联合工作解决了 Coxeter-Catalan 组合学中长达二十年的开放问题，同时产生了有理非交叉 Coxeter-Catalan 对象的第一个定义，同时还给出了非交叉对象的第一个统一枚举。 在处理复数时，也可以与麦克唐纳理论（对角调和和 q,t 组合）联系起来。 在不同的通用性级别上，可以使用不同的技术。 对于有限 Coxeter 群，可以使用 Hecke 代数的显式分解以具体情况计算所有内容，并且有许多有趣的组合和表示理论问题可供立即攻击。 有限类型中的特殊类型的元素具有有利的表示理论属性，允许统一的方法。 对于仿射 Weyl 群，主要工具是由 Opdam 提供的平移迹公式。 例如，所提出的框架在这​​种情况下恢复了 Haglund 的一些 Tessler 矩阵恒等式。 对于一般的 Kac-Moody Weyl 群，我们被简化为一般的递归和聚类理论方法。 这些方法也适用于有限和仿射情况，但在进一步探索之前需要软件实现。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。