Combinatorial Representation Theory of Quantum Groups and Coinvariant Algebras

组合学被描述为数学的纳米技术。它涉及到对离散对象进行计数，这在许多应用程序中自然会出现。例如，软件开发经常需要在不同的算法之间进行选择来解决问题。组合数学允许人们计算每个候选算法所走的步数，然后选择最佳解决方案。通过这种方式，组合学提供了一套基本工具和一组论点原型，指导整个STEM中问题的解决。组合数学研究的优点之一是它为学生提供了发展问题解决、软件开发和其他关键技能的具体机会。更具体地说，代数组合数学专注于来自拓扑学、表示论、粒子物理和其他领域的高度结构化的、通常是高级问题的组合本质。这样的问题经常以某种方式简化为复杂的组合分析。一个这样的代数问题就是理解量子群。这些非凡的结构产生于1980年左右，与量子力学中的可积晶格模型有关，而纯数学和物理学中一些技术上最深刻的理论就是在这个领域。本项目的主要重点之一是进一步开发称为Web库的某些组合图。这些组合对象编码了量子群的表示范畴，并允许利用强大的拓扑量子不变量进行高效的计算。它们连接了一个非常不同的主题集合，包括全正性、交替符号矩阵、平面划分、晶基、动态代数组合学和仿射Grassmanian的几何。学生将参与研究项目。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Combinatorics has been described as the nanotechnology of mathematics. It is concerned with counting discrete objects, which naturally arise in many applications. As one example, software development frequently requires choosing between different algorithms to solve a problem. Combinatorics allows one to count the number of steps each candidate algorithm takes and then choose the best solution. In this way, combinatorics provides a set of basic tools and a collection of argument prototypes that guide the solution of problems throughout STEM. One of the virtues of combinatorics research is that it provides students with concrete opportunities to develop problem-solving, software development, and other key skills.Algebraic combinatorics, more specifically, focuses on the combinatorial essence of highly structured and often advanced problems coming from topology, representation theory, particle physics, and other areas. Such problems are frequently reduced in some fashion to an intricate combinatorial analysis. One such algebraic problem is to understand quantum groups. These remarkable structures arose around 1980 from connections with integrable lattice models in quantum mechanics, and some of the technically deepest theories in pure mathematics and physics are in this area. One of the main focuses of the present project is to further develop certain combinatorial diagrams called web bases. These combinatorial objects encode the representation category of quantum groups and allow for efficient computations with powerful topological quantum invariants. They connect a remarkably diverse collection of topics, including total positivity, alternating sign matrices, plane partitions, crystal bases, dynamical algebraic combinatorics, and the geometry of the affine Grassmannian. Students will be involved in the research project.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.