Combinatorics and Braid Varieties

代数组合学是数学的一个分支，它用组合方法研究代数结构，用代数方法研究组合结构。 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背景下-代数组合学因此在密码学、蛋白质折叠、高能物理和量子计算等学科中有着不同的应用。 这个项目将使用代数对象和方法来产生新的组合结果，利用辫子品种-一种与结相关的代数空间-作为统一工具。 资金将额外支持培训研究生和推广工作，包括工作的互动在线离散数学textbooks.In更详细的，这个建议提出了一个框架，用于生产组合的结果，使用辫子品种在有限域，Hecke代数迹，合理的Cherednik代数，和一个新的关系与noncrossing组合。 该框架已经被证明成功地产生了大量的新结果：PI最近与Galashin，Lam和Trinh的联合工作解决了Coxeter-Catalan组合学中长达20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--对角谐波和q，t组合学--在处理复数时也是预期的。 在不同的一般性水平上，可以使用不同的技术。 对于有限Coxeter群，可以使用Hecke代数的显式分解以逐个情况的方式计算所有内容，并且有许多有趣的组合和表示论问题可以立即攻击。 有限型元素的特殊类别具有良好的表示理论性质，允许统一的方法。 对于仿射外尔群，主要的工具是一个平移的迹公式，这是由Opdam提出的。 例如，建议的框架恢复一些Tessler矩阵身份由于Haglund在此设置。 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的情况下，但将需要软件实现之前，进一步exploration.This奖项反映了NSF的法定使命，并已被认为是值得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的支持。

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.