Polyhedral Combinatorics in Representation Theory and Algebraic Geometry

表示论和代数几何中的多面体组合

• 批准号: 0500534
• 负责人: Andrei Zelevinsky
• 金额: --
• 依托单位: Northeastern University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0500534&HistoricalAwards=false
• 关键词: Polyhedral; Combinatorics; Representation; Theory; Algebraic

The project focuses on cluster algebras discovered by theinvestigator in collaboration with S.Fomin. Cluster algebras arecommutative of a special kind designed to provide an algebraicframework for the study of total positivity and canonical bases insemisimple groups and their representations. The investigator studiesstructural properties of cluster algebras and their quantumdeformations. This study uncovers unexpected connections with suchdiverse subjects as the structural theory of Kac-Moody algebras,representation theory of finite dimensional algebras, geometry ofmoduli spaces, and the theory of superpotentials. One of the maininstruments of the study is polyhedral combinatorics. This project is motivated by two classical areas of mathematics:representation theory and the theory of total positivity.Representation theory is a mathematical approach to studying symmetry;more specifically, it encodes the symmetry properties of variousphysical and biological systems that occur in nature. Total positivityis a remarkable property of matrices (square arrays of numbers) thatgeneralizes the familiar notion of positive numbers. Both theoriesfind numerous applications in physics, chemistry and other sciences,as well as in other mathematical disciplines. In fact, representationtheory serves as the mathematical foundation of quantum mechanics,while total positivity is a major tool for explaining oscillations inmechanical systems. During the last decade, deep connections werefound between the two fields, and the scope of their applications wasgreatly extended. This project explores the modern framework ofrepresentation theory and total positivity, with the goal of makingits formalism more explicit and understandable.

该项目重点关注研究人员与 S.Fomin 合作发现的簇代数。簇代数是一种特殊类型的交换，旨在为研究半单群及其表示中的总正性和规范基提供代数框架。研究人员研究簇代数的结构性质及其量子变形。这项研究揭示了与 Kac-Moody 代数结构理论、有限维代数表示论、模空间几何和超势理论等不同学科的意想不到的联系。该研究的主要工具之一是多面体组合学。 该项目的动机是数学的两个经典领域：表示论和完全积极性理论。表示论是一种研究对称性的数学方法；更具体地说，它编码了自然界中发生的各种物理和生物系统的对称性。总正性是矩阵（数字的方阵）的一个显着属性，它概括了熟悉的正数概念。这两种理论在物理、化学和其他科学以及其他数学学科中都有广泛的应用。事实上，表示论是量子力学的数学基础，而完全实证性是解释机械系统振荡的主要工具。在过去的十年中，这两个领域之间建立了深厚的联系，其应用范围也大大扩展。该项目探索了表征理论和完全实证性的现代框架，其目标是使其形式主义更加明确和易于理解。