Combinatorial Representation Theory of Quantum Groups and Coinvariant Algebras

量子群与协变代数的组合表示论

• 批准号: 2348843
• 负责人: Joshua Swanson
• 金额: $ 18万
• 依托单位: University of Southern California
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-07-01 至 2027-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2348843&HistoricalAwards=false
• 关键词: Combinatorial; Representation; Theory; Quantum; Groups

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.

组合学被描述为数学的纳米技术。它与计算离散对象有关，这些对象自然在许多应用中出现。作为一个例子，软件开发通常需要在不同的算法之间选择解决问题。 Compinatorics允许一个人计算每个候选算法所需的步骤数，然后选择最佳解决方案。这样，Combinatorics提供了一组基本工具和一系列参数原型，可指导整个STEM的问题解决方案。组合学研究的优点之一是，它为学生提供了开发问题解决问题，软件开发和其他关键技能的具体机会。更具体地说的是，核心组合学，侧重于高度结构化和经常来自拓扑，代表理论，粒子和其他领域的高度结构化和经常先进问题的组合本质。这种问题通常以某种方式减少到复杂的组合分析。这样的代数问题是了解量子群。这些出色的结构左右左右来自与量子力学中的可整合晶格模型的联系，而纯数学和物理学中的一些技术最深的理论都在该领域。本项目的主要重点之一是进一步开发某些称为Web基础的组合图。这些组合对象编码量子组的表示类别，并允许使用功能强大的拓扑量子不变性的有效计算。他们连接了一个非常多样化的主题集合，包括总阳性，交替的符号矩阵，平面隔板，晶体底座，动力学代数组合和仿射grassmannian的几何形状。学生将参与研究项目。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的评论标准来评估值得支持的。