Algebraic Combinatorics

代数组合学

• 批准号: 0245385
• 负责人: John Stembridge
• 金额: --
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0245385&HistoricalAwards=false
• 关键词: Algebraic; Combinatorics

Abstract for the proposal DMS- 0245385 of Stembridge and FominThe investigators will continue their study of combinatorialstructures arising in algebra, representation theory and algebraicgeometry, and to develop software tools to assist researchers inthese areas. One example of such structures are the cluster algebras,designed to provide a concrete algebraic framework for dual canonicalbases and total positivity in semisimple groups. Our investigationswill also include the study of generalized Kostka-Foulkes polynomials,sub-Coxeter arrangements, and the combinatorics of crystal graphs andgeneralized associahedra.In selecting combinatorial problems and conjectures to investigate,the proposers intent has been to identify areas that are most likely toprovide insight into other disciplines. Indeed, combinatorics has animportant role to play in an ever widening array of applications inboth mathematics and computer science; this increasing importance isrelated to the demand for explicit or algorithmic understanding ofdiscrete structures, and has grown in tandem with the prominence ofcomputation in nearly all phases of research in mathematics.

对于Stembridge和Fomin的DMS-0245385建议，研究人员将继续研究代数、表示理论和代数几何中出现的组合结构，并开发软件工具来帮助这些领域的研究人员。这种结构的一个例子是簇代数，它被设计成为半单群中的对偶正则基和全正性提供一个具体的代数框架。我们的研究还将包括广义Kostka-Foulkes多项式的研究，次Coxeter排列的研究，以及结晶图和广义结合图的组合学的研究。在选择组合问题和猜想进行研究时，提出者的意图是确定最有可能为其他学科提供见解的领域。事实上，组合学在数学和计算机科学中不断扩大的应用范围中发挥着重要的作用；这种日益增长的重要性与对离散结构的显式或算法理解的需求有关，并随着计算在数学研究的几乎所有阶段的突出而增长。