Applications of Quiver Representations to Algebra and Geometry

项目负责人：Jerzy Weyman项目编号：0300064单位：东北大学摘要：抖动表示在代数和几何中的应用。提案由几个相互关联的部分组成。第一部分是研究颤子的半不变量环及其所定义的组合不变量。研究者建议继续研究半不变量环权的锥壁和这些环权空间的多重性。在一种特殊情况下，这包括由Klyachko不等式定义的锥。这一部分还包括对颤振表示的协变模的研究。提出了计算Dynkin颤振维数缺陷的方法。第二部分讨论带关系颤振的半不变量。这里的主要问题是用半不变量来描述那些具有有限类型的弱颤动。第三部分是研究与约化群有关的广义颤振。特别地，研究者提出研究对称颤振的半不变量环。他的研究生史蒂夫·洛维特研究了这类颤振的轨道闭合。最后一部分研究了余维三、四振子和对称振子的轨道闭包的定义理想如何与余维三理想和余维四理想的结构理论联系起来。非技术描述。这一建议涉及代数的两个分支：颤子的表示和交换代数。抖动的表示是一种将矢量数据与一些有向图的顶点相关联的方法。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些对象。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，研究者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。该建议的最后一部分将由这些方程（co维四的Gorenstein理想）定义的某些类型的对象与颤振的表示联系起来。如果成功，这将导致对这些对象的组合描述。

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.