Geometric aspects of quiver representations

DMS-0600229Jerzy M. PI建议研究代数的代数表示和其他分支之间的相互作用。提出了几个问题：研究箭图半不变量环的权重锥壁及其权重空间的多重性，用半不变量来表征有限型和驯服箭图，余维3和余维4的Dynkin箭图的轨道闭包与余维3的完全理想和余维Gorenstein理想的结构之间的预期联系四涉及集群代数的部分包括研究与Igusa-Orr的图像理论的联系以及研究具有超势的箭图的突变。该提议涉及箭图的表示、交换代数和表示论-代数的不同分支之间的相互作用。箭图的表示为编码线性代数问题，即涉及向量和矩阵的问题提供了一个很好的组合工具。交换代数和代数几何是研究由多项式方程定义的集合的数学分支。PI建议研究的一些相互作用是最近发现的，并为所有这些领域提供了新的见解。在本提案中，可能的新的未预料到的联系之间的颤抖和结构的完善和Gorenstein理想将被探讨。如果这一部分实现了，它可能会在交换代数中产生根本性的影响。

DMS-0600229Jerzy M. WeymanThe PI proposes to study interactions between the quiver representations and other branches of algebra. Several problems are proposed: study of the walls of cones of weights of rings of semi-invariants of quivers and the multiplicities of their weight spaces, characterizing finite type and tame quivers in terms of semi-invariants, connections between quiver representations and cluster algebras and an expected connection between the orbit closures for Dynkin quivers of codimension three and four and the structure of perfect ideals of codimension three and Gorenstein ideals of codimension four. The part involving cluster algebras includes studying connections with the theory of pictures of Igusa-Orr and studying the mutations of quivers with superpotential.This proposal is concerned with interactions between the representations of quivers, commutative algebra and representation theory - different branches of algebra. The representations of quivers give a nice combinatorial tool for coding the linear algebra problems, i.e. problems involving vectors and matrices. Commutative algebra and algebraic geometry are branches of mathematics studying the sets defined by polynomial equations. Some of the interactions the PI proposes to study were discovered recently and provided new insights to all these areas. In the present proposal the possible new unanticipated connection between quivers and the structure of perfect and Gorenstein ideals will be explored. If this part materializes, it could have fundamental consequences in commutative algebra.