Computational Invariant Theory

In the fall of 1997 I gave a series of lectures at Queen’s University on algorithms in invariant theory of finite groups. This article is an expanded version of the material presented there. The main topic is the calculation of the invariant ring of a finite group acting on a polynomial ring by linear transformations of the indeterminates. By “calculation” I mean finding a finite system of generators for the invariant ring, and (optionally) determining structural properties of it. In this exposition particular emphasis is placed on the case that the ground field has positive characteristic dividing the group order. We call this the modular case, and it is important for several reasons. First, many theoretical questions about the structure of modular invariant rings are still open. I will address the problems which I consider the most important or fascinating in the course of the paper. Thus it is very helpful to be able to compute modular invariant rings in order to gain experience, formulate or check conjectures, and gather some insight which in fortunate cases leads to proofs. Furthermore, the computation of modular invariant ring can be very useful for the study of cohomology of finite groups (see Adem and Milgram [1]). This exposition also treats the nonmodular case (characteristic zero or coprime to the group order), where computations are much easier and the theory is for the most part settled. There are also various applications in this case, such as the solution of algebraic equations or the study of dynamical systems with symmetries (see, for example, Gatermann [11], Worfolk [26]). This is not a research paper, and so there is no claim of originality. In fact, most of the material is covered by the papers [14,15,18]. The goal is to give a coherent exposition of what is scattered through several original papers, which I hope is readable and assumes as little knowledge as possible. There are several implementations of the algorithms treated in this text. The most efficient of these is contained in the Magma system (see Kemper and Steel [18] and Bosma et al. [6]). There is an older implementation in Maple written by myself, which can be obtained by anonymous ftp from the site ftp.iwr.uni-heidelberg.de under /pub/kemper/INVAR2. A further implementation in Singular has been written by Agnes Heydtmann (email [email protected]).