Applications of Quiver Representations to Algebra and Geometry

Quiver 表示在代数和几何中的应用

• 批准号: 0300064
• 负责人: Jerzy Weyman
• 金额: $ 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: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0300064&HistoricalAwards=false
• 关键词: Applications; Quiver; Representations; Algebra; Geometry

Principal Investigator: Jerzy Weyman Proposal Number: 0300064Institution: Northeastern UniversityAbstract: Applications of Quiver Representations to Algebra and GeometryTechnical description. The proposal consists of several interrelated parts. The first part is to study the rings of semi-invariants of quivers and the combinatorial invariants they define. The investigator proposes to continue to study the walls of cones of weights of rings of semiinvariants and the multiplicities of weight spaces for these rings. In one particular case this includes the cones defined by Klyachko inequalities. This part also includes the study of modules of covariants for quiver representations. The investigator proposes to compute the defect of the dimension vectors for the Dynkin quivers. The second part is concerned with semi-invariants for quivers with relations. The main problem here is to characterize in terms of semi-invariants those quivers with relations that are tame and have finite type. The third part is to study the generalized quivers associated to reductive groups. In particular the investigator proposes to study the rings of semi-invariants of symmetric quivers. His graduate student Steve Lovett studies the orbit closures for such quivers. The last part consists of studying how the defining ideals of orbit closures of codimension three and four for quivers and symmetric quivers are connected to the structure theory of perfect ideals of codimension three and Gorenstein ideals of codimension four.Non-technical description.This proposal is related to two branches of algebra: representations of quivers and commutative algebra. A representation of a quiver is a way to associate vector data to thevertices of some oriented graph. The edges of a graph can be viewed as relations between these data. Abstract algebra allows us to study such objects systematically. The results of this research might lead to better algorithms for dealing with linear algebra problems. In fact some of the published research of the investigator has led to such algorithms. Commutative algebra studies sets defined by polynomial equations. The last part of the proposal relates certain types of objects defined by such equations (Gorenstein ideals of codimension four) to representations of quivers. If successful this would lead to a combinatorial description of such objects.

主要研究人员：Jerzy Weyman建议编号：0300064机构：东北大学摘要：箭图表示在代数和几何技术描述中的应用。该提案由几个相互关联的部分组成。第一部分研究箭图的半不变量环及其定义的组合不变量。作者建议继续研究半不变量环的权的锥壁和这些环的权空间的重数。在一种特殊情况下，这包括由Klyachko不等式定义的锥体。这一部分还包括箭图表示的协变量模的研究。研究人员建议计算动态抖动的维矢量的亏损值。第二部分是带关系箭图的半不变量。这里的主要问题是用半不变量来刻画那些带有温和和有限类型关系的箭图。第三部分研究了与约化群相关的广义箭图。特别是，研究者建议研究对称箭图的半不变量环。他的研究生史蒂夫·洛维特研究这种抖动的轨道闭合现象。最后一部分研究了关于箭图和对称箭图的余维3和4的轨道闭包的定义理想如何与余维3的完全理想和余维4的Gorenstein理想的结构理论相联系。箭图的表示是将向量数据与某个有向图的顶点相关联的一种方式。图的边可以看作是这些数据之间的关系。抽象代数使我们能够系统地研究这样的对象。这项研究的结果可能会导致更好的算法来处理线性代数问题。事实上，这位研究人员发表的一些研究已经导致了这样的算法。交换代数研究由多项式方程定义的集合。提案的最后一部分将由此类方程定义的某些类型的对象(余维为四的Gorenstein理想)与箭图的表示联系起来。如果成功，这将导致对这类物体的组合描述。