Combinatorial Representation Theory of Quantum Groups and Coinvariant Algebras

组合学被描述为数学的纳米技术。它关注的是离散对象的计数，这在许多应用中自然会出现。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。组合学允许计算每个候选算法所采取的步骤数，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参数原型，指导整个STEM问题的解决。组合研究的优点之一是它为学生提供了发展解决问题、软件开发和其他关键技能的具体机会。更具体地说，代数组合学侧重于高度结构化的组合本质，通常是来自拓扑学、表示理论、粒子物理和其他领域的高级问题。这类问题常常以某种方式简化为复杂的组合分析。其中一个代数问题就是如何理解量子群。这些引人注目的结构出现在1980年左右，与量子力学中的可积晶格模型有关，纯数学和物理学中一些技术上最深奥的理论都在这个领域。本项目的主要焦点之一是进一步开发某些称为web库的组合图。这些组合对象对量子群的表示类别进行编码，并允许使用强大的拓扑量子不变量进行高效计算。它们连接了一个非常不同的主题集合，包括总正性、交替符号矩阵、平面分区、晶体基、动态代数组合学和仿射格拉斯曼几何。学生将参与研究项目。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。

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.