Combinatorics and Braid Varieties

代数组合学是用组合方法研究代数结构和用代数方法研究组合结构的数学分支。这样的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有不同的应用。该项目将使用代数对象和方法来产生新的组合结果，利用辫变体（一种与结相关的代数空间）作为统一工具。资金还将用于培训研究生和拓展工作，包括编写交互式在线离散数学教科书。更详细地说，这个建议提出了一个利用有限域上的辫状变异、Hecke代数迹、理性Cherednik代数以及与非交叉组合的新关系来产生组合结果的框架。这个框架已经被证明在产生实质性的新结果方面是成功的：PI最近与Galashin， Lam和Trinh的联合工作解决了coxet - catalan组合学中长达二十年的开放问题，同时产生了第一个理性非交叉coxet - catalan对象的定义，同时也给出了第一个非交叉对象的统一枚举。与麦克唐纳理论的联系——对角线谐波和q，t组合——在处理复数时也被期望。在不同的通用性层次上，可以使用不同的技术。对于有限的Coxeter群，可以使用Hecke代数的显式分解逐个计算所有内容，并且有许多有趣的组合和表示理论问题可供立即攻击。有限型元素的特殊类别具有有利的表示理论性质，允许统一的方法。对于仿射Weyl群，主要工具是用于翻译的跟踪公式，这要归功于Opdam。例如，在这种情况下，由于Haglund，所提出的框架恢复了一些Tessler矩阵恒等式。对于一般的Kac-Moody Weyl群，我们将其简化为一般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。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.