Applications of Quiver Representations to Algebra and Geometry

主要研究者：Jerzy Weyman提案编号：0300064机构：东北大学摘要：箭图表示在代数和几何中的应用技术描述。该提案由几个相互关联的部分组成。第一部分是研究箭图的半不变量环及其定义的组合不变量。研究者建议继续研究半不变环的权锥的壁和这些环的权空间的多重性。在一个特殊的情况下，这包括由Klyachko不等式定义的锥。这一部分也包括了对协变模的研究。研究者提出了计算Dynkin抖动器的维数向量的缺陷。第二部分是关于有关系的箭图的半不变量。这里的主要问题是描述的半不变量，这些颤抖的关系是驯服的，并有有限的类型。第三部分是研究与约化群相关的广义箭图。特别是调查人员提出研究环的半不变量的对称颤动。他的研究生史蒂夫·洛维特（Steve Lovett）研究了这种颤动的轨道闭合。最后一部分研究了余维3和4轨道闭包的定义理想与余维3和余维4 Gorenstein理想的结构理论之间的关系，这一关系涉及代数的两个分支：余维表示和交换代数。一个向量的表示是一种将向量数据与某个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，一些研究人员发表的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涉及某些类型的对象定义的方程（Gorenstein理想的余维4）表示的箭袋。如果成功的话，这将导致对这些物体的组合描述。

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.