Higher Representations and Derived Equivalences

更高的表示和派生等价

• 批准号: 1702305
• 负责人: Raphael Rouquier
• 金额: $ 55万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1702305&HistoricalAwards=false
• 关键词: Higher; Representations; Derived; Equivalences

This project will further the understanding of symmetries as a fundamental tool to organize structures in mathematics. The PI will bring new methods and results in representation theory, which is the mathematical study of symmetries via linear algebra, using higher categorical and homotopical methods. In the process, the PI will develop new computational approaches and new combinatorial objects are expected to arise. This project will topologically enhance classical structures in algebra in order to analyze symmetries and fixed points, with the ultimate aim of resolving some fundamental questions in representation theory. A key goal of the project is to make progress in the understanding of simple modules and decomposition matrices of finite groups of Lie type using perverse equivalences. The PI will work toward a proof of Broue's abelian defect group conjecture via Deligne-Lusztig varieties. This will require developing new homotopical methods to deal with certain fixed point constructions. The PI will also investigate genericity properties and look for combinatorial objects to encode numerical information.

该项目将进一步了解对称性作为组织数学结构的基本工具。PI将带来新的方法和结果表示论，这是通过线性代数的对称性的数学研究，使用更高的分类和同伦方法。在这个过程中，PI将开发新的计算方法，预计将出现新的组合对象。这个项目将在拓扑上增强代数中的经典结构，以分析对称性和不动点，最终目的是解决表示论中的一些基本问题。该项目的一个关键目标是在理解李型有限群的简单模块和分解矩阵方面取得进展。PI将致力于通过Deligne-Lusztig变种证明Broue的阿贝尔亏群猜想。这就需要发展新的同伦方法来处理某些不动点的构造。PI还将研究泛型属性，并寻找组合对象来编码数字信息。

项目成果

Brauer trees of unipotent blocks

• DOI: 10.4171/jems/978
• 发表时间: 2020-06
• 期刊: Journal of the European Mathematical Society
• 影响因子: 2.6
• 作者: David A. Craven;O. Dudas;Raphaël Rouquier