Applications of Quiver Representations to Algebra and Geometry

Northeastern University, Boston MA

Investigators

Abstract

Principal Investigator: Jerzy Weyman Proposal Number: 0300064 Institution: Northeastern University Abstract: Applications of Quiver Representations to Algebra and Geometry Technical 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 the vertices 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.