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: 0200299
• 负责人: Andrei Zelevinsky
• 金额: $ 16.12万
• 依托单位: Northeastern University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-05-01 至 2006-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0200299&HistoricalAwards=false
• 关键词: Polyhedral; Combinatorics; Representation; Theory; Algebraic

The project focuses on cluster algebras recently discovered by theinvestigator in collaboration with S.Fomin. Cluster algebras areintegral domains of a special kind designed to provide analgebraic framework for the study of total positivity andcanonical bases in semisimple groups and their representations.The investigator studies structural properties of cluster algebrasand their quantum deformations. This study uncovers unexpectedconnections with such diverse subjects as the structural theory ofKac-Moody algebras, thermodynamic Bethe ansatz, quiverrepresentations, and integrable systems. One of the maininstruments of the study is polyhedral combinatorics, morespecifically, an interplay between piecewise-linear andsubtraction-free birational transformations based on the tropicalcalculus.The main motivation for this project comes from two classical areasof mathematics: representation theory and the theory of totalpositivity. Representation theory is a mathematical approach tostudying symmetry; more specifically, it encodes the symmetryproperties of various physical and biological systems that occur innature. Total positivity is a remarkable property of matrices (arraysof numbers) that generalizes the familiar notion of positive numbers.Both theoriesfind numerous applications in physics, chemistry and othersciences, as well as in other mathematical disciplines. In fact,representation theory serves as the mathematical foundation ofquantum mechanics, while total positivity is a major tool forexplaining oscillations in mechanical systems. During the lastdecade, deep connections were found between the two fields, andthe scope of their applications was greatly extended. Thisproject explores the modern framework of representationtheory and total positivity, with the goal of making its formalismmuch more explicit and understandable.

该项目的重点是最近由研究者与S.Fomin合作发现的簇代数。聚类代数是一类特殊的积分域，为研究半单群中的全正基和正则基及其表示提供了代数框架。研究者研究了簇代数的结构性质及其量子变形。本研究揭示了与kac - moody代数的结构理论、热力学贝特解、振动表示和可积系统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，更具体地说，是基于热带微积分的分段线性和无减法两种变换之间的相互作用。这个项目的主要动机来自数学的两个经典领域：表示理论和总体正性理论。表征理论是研究对称性的一种数学方法；更具体地说，它编码了自然界中各种物理和生物系统的对称性。总正性是矩阵（数组或数字）的一个显著性质，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，表征理论是量子力学的数学基础，而总正性是解释力学系统振荡的主要工具。在过去的十年中，这两个领域之间发现了深刻的联系，它们的应用范围大大扩大了。该项目探索了表征理论和整体积极性的现代框架，目的是使其形式主义更加明确和可理解。