Applications of Representations of Quivers

摘要。这一提议是关于颤振的表示，几何和表示理论之间的相互作用。研究者将研究颤子的半不变量环及其所定义的组合不变量。我们将特别注意这种环的权锥及其与表示理论的关系。在三旗抖动的特殊情况下，这相当于研究由Klyachko不等式定义的锥。主要研究者还将研究与约化群及其半不变量相关的广义颤振，以及具有有限多轨道的齐次空间的积。箭袋只是有向图。它们的表示提供了一种方便的编码方案，允许研究在数学的各个领域中出现的对象分类。矩阵的秩分类和向量空间自同态的约旦分类是最简单的例子。更复杂的问题包括经典群的表示及其对无限维代数的推广。研究这类分类问题，以及它们可以明确解决的条件，是数学研究的核心。研究者和他的合作者发现，使用不变理论的方法可以获得新的有趣的结果，而这些方法在以前被忽视了。

Abstract.This proposal is concerned with interactions between the representations of quivers, geometry and representation theory.The investigator will study the rings of semi-invariants of quivers and the combinatorial invariants they define. Special attention will be paid to the cone of weights of such rings and its relation to representation theory. In the special case of triple flag quivers this amounts to studying the cone defined by Klyachko inequalities. The principal investigator will also study the generalized quivers associated to reductive groups and their semi-invariants as well as the products of homogeneous spaces with finitely many orbits.Quivers are just oriented graphs. Their representations provide a convenient coding scheme allowing to study classification of objects arising in various areas of mathematics. The classification of matrices by their ranks and Jordan classificaton of endomorphisms of a vector space are the simplest examples. The more complicated ones include representations of classical groups and their generalizations to infinite dimensional algebras. Studying such classification problems, and conditions under which they can be explicitly solved is central in mathematical research. The investigator and his collaborators found that new interesting results can be obtained by using methods of invariant theory which were neglected before.