Quivers, Invariant Theory and Applications

箭袋、不变理论及其应用

• 批准号: 0102193
• 负责人: Harm Derksen
• 金额: $ 9.55万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-07-15 至 2005-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0102193&HistoricalAwards=false
• 关键词: Quivers; Invariant; Theory; Applications

The investigator works on Invariant Theory and representations of quivers. He studies some fundamental problems in the theory of quiver representations. In earlier joint work with Jerzy Weyman, he found a nice description and even an algorithm for the decomposition of a general representation into indecomposable representations (this is the canonical decomposition as introduced by Kac). Also Weyman and the investigator proved that semi-invariants introduced by Schofield always generate the ring of semi-invariants. The above results have a remarkable application to Littlewood-Richardson coeffients. In particular one can prove results of Klyachko, and Knutson-Tao about the set of nonzero Littlewood-Richardson coefficients using quiver representations. Using results about quiver representations many more new results about Littlewood-Richardson coefficients can be proven. The investigator is continuing his research on quiver representations to deepen our understanding of generic representations of wild quivers, and to apply these results to the combinatorics of Littlewood-Richardson coefficients. The investigator just finished writing a book together with Gregor Kemper on Computational Invariant Theory. He is now studying various problems in Invariant Theory, in particular algorithms for finding rational invariants and for determining whether two elements in a given representation of an algebraic group lie in the same orbit. This orbit problem is the original motivation of Invariant Theory.This research is in the area of Mathematics referred to as Invariant Theory, a branch of Representation Theory, in the general area of Algebra. Invariant Theory has a long tradition back to the nineteenth century. It studies quantities which stay invariant under certain symmetries (in arbitrary dimension). A simple example is a person on earth. The quantity "distance to the rotation axes" stays invariant under rotation of the earth. Of course symmetries play an important role in nature. Related to this is the problem to recognize if two objects can be transformed into each other by certain symmetries. Think of a robot eye which has to recognize whether two objects are the same after rotation. The investigator studies various problems in invariant theory and in particular (a mathematical formulated version of) the object recognition problem. "Quiver" representation theory can be thought of as a generalization of linear algebra. This theory shows a deep and interesting structure. For example, a graphical representation gives fractal-like pictures. There are several applications to other branches of mathematics.

研究者研究不变量理论和颤振的表示。他研究了颤振表示理论中的一些基本问题。在早期与Jerzy Weyman的合作中，他发现了一个很好的描述，甚至是一个将一般表示分解为不可分解表示的算法（这是由Kac引入的规范分解）。Weyman等人还证明了Schofield引入的半不变量总是生成半不变量环。上述结果对Littlewood-Richardson系数具有显著的应用价值。特别是可以用颤振表示证明Klyachko和Knutson-Tao关于非零Littlewood-Richardson系数集的结果。利用颤振表示的结果可以证明更多关于Littlewood-Richardson系数的新结果。研究者正在继续他对颤振表示的研究，以加深我们对野生颤振的一般表示的理解，并将这些结果应用于Littlewood-Richardson系数的组合学。这位研究者刚刚和格雷戈尔·肯珀一起写了一本关于计算不变量理论的书。他现在正在研究不变量理论中的各种问题，特别是寻找有理不变量的算法，以及确定给定代数群表示中的两个元素是否在同一轨道上的算法。这个轨道问题是不变量理论的原始动机。这项研究是在数学领域被称为不变理论，表示理论的一个分支，在代数的一般领域。不变量理论有着悠久的传统，可以追溯到19世纪。它研究在一定对称（任意维）下保持不变的量。一个简单的例子是地球上的一个人。“到旋转轴的距离”这个量在地球自转下保持不变。当然，对称性在自然界中扮演着重要的角色。与此相关的问题是，如何识别两个物体是否可以通过某些对称性相互转化。想象一下机器人的眼睛，它必须在旋转后识别两个物体是否相同。研究者研究了不变量理论中的各种问题，特别是（数学形式的）对象识别问题。“颤抖”表示理论可以被认为是线性代数的推广。这个理论显示了一个深刻而有趣的结构。例如，图形表示给出了类似分形的图像。在数学的其他分支中也有一些应用。