Quivers, Invariant Theory and Applications

这位研究者致力于不变理论和箭图的表示。他研究箭图表象理论中的一些基本问题。在与Jerzy Weyman的早期合作中，他发现了一种很好的描述，甚至找到了一种将一般表示分解为不可分解表示的算法(这是Kac引入的规范分解)。Weyman和研究者还证明了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。特别地，可以使用箭图表示证明Klyachko和Knutson-Tao关于非零Littlewood-Richardson系数集的结果。利用箭图表示的结果，可以证明更多关于Littlewood-Richardson系数的新结果。这位研究者正在继续他对箭图表示的研究，以加深我们对野生箭图的一般表示的理解，并将这些结果应用于Littlewood-Richardson系数的组合学。这位研究人员刚刚与格雷戈尔·肯珀一起完成了一本关于计算不变量理论的书。他现在正在研究不变理论中的各种问题，特别是寻找有理不变量和确定代数群的给定表示中的两个元素是否位于同一轨道上的算法。这个轨道问题是不变理论产生的原始动机。这项研究是在数学领域内进行的，被称为不变理论，它是表示论的一个分支，在一般的代数领域。不变量理论有着悠久的传统，可以追溯到19世纪。它研究在一定的对称性下(在任意维度)保持不变的量。一个简单的例子是地球上的一个人。在地球自转的情况下，“到自转轴的距离”这个量是不变的。当然，对称性在自然界中扮演着重要的角色。与此相关的是识别两个对象是否可以通过某些对称相互转换的问题。想想机器眼，它必须识别两个物体在旋转后是否相同。研究人员研究不变量理论中的各种问题，特别是物体识别问题(一个数学形式的版本)。“箭图”表示理论可以看作是线性代数的推广。这一理论展现了一种深刻而有趣的结构。例如，图形表示提供了类似于分形图的图片。在数学的其他分支中也有几个应用。

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.