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.
