Quivers, Invariant Theory and Applications

研究人员致力于不变量理论和颤抖的表示。他研究了一些基本问题的理论的代表性。 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Kac引入的规范分解）。 此外，韦曼和调查证明，半不变量介绍了斯科菲尔德总是产生环的半不变量。上述结果对Littlewood-Richardson系数有显著的应用。 特别是可以证明结果Klyachko，和Knutson-Tao关于一组非零Littlewood-Richardson系数使用的表示。 利用这些结果可以证明更多关于Littlewood-Richardson系数的新结果。 研究人员正在继续他的研究，对numerical表示，以加深我们的理解，通用表示野生颤抖，并将这些结果应用到组合的Littlewood-Richardson系数。调查员刚和格雷戈尔·肯珀一起写了一本关于计算不变理论的书。 他现在正在研究各种问题的不变量理论，特别是算法寻找合理的不变量，并确定是否有两个元素在一个给定的代表性的代数群位于同一轨道。 这个轨道问题是不变论的最初动机，这个研究是在数学领域中被称为不变论的，它是代数领域中表示论的一个分支。 不变量理论有着悠久的传统，可以追溯到世纪。 它研究在某些对称性下保持不变的量（在任意维度上）。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。 与此相关的问题是认识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，它必须识别两个物体在旋转后是否相同。 调查研究各种问题的不变量理论，特别是（数学制定版本）的对象识别问题。“箭图”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片。 它在数学的其他分支中也有一些应用。

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.