1. First Steps

Abstract

Invariant theory seeks to determine whether a (mathematical) object can be obtained from some other object by the action of some group. One way to answer this question is to find some functions that map from the class of objects to some field (or more generally some ring). Invariants are functions which take the same value on any two objects which are related by an element of the group. Thus if we can find any invariant which takes different values on two objects, then these two objects cannot be related by an element of the group. Ideally, we hope to find enough invariants to separate all objects which are not related by any group element. This means we want to find a (finite) set of invariants f1, f2, . . . , fr with the property that if two objects are not related by the group action then at least one of these r invariants takes different values on the two objects in question. For example, suppose we wish to determine whether two triangles are congruent, that is, whether one can be obtained from the other by translation, rotation, reflection or a combination of these operations. One useful invariant is the area function: two triangles with different areas cannot be congruent. On the other hand, the (unordered) set of three functions which give the lengths of the three sides are sufficient: two different triangles having sides of the same lengths must be congruent. For us, the mathematical objects are elements of some vector space with a group action and the invariants will be those regular functions on the vector space that are constant on each of the group orbits. We begin with some basic material on the action of groups on vector spaces and their coordinate rings, followed by a simple illustrative example. There are excellent references available: Benson [6], Derksen and Kemper [26], Neusel [85], Neusel and Smith [86] and Smith [103]. We also note that many advances in modular invariant theory have been made due to the programming language MAGMA [10], especially the invariant theory packages developed by Gregor Kemper.