Algorithmic invariant theory

Abstract

Invariant theory can be put in a very general context: If “∼” is an equivalence relation on a set X, then an invariant is a function on X which is constant on every equivalence class. So invariants serve to parametrize equivalence classes. The goals of invariant theory are to find all invariants that meet some further restrictions (such as continuity or polynomiality), and to study to which extent these invariants separate equivalence classes. For example, the determinant of a square matrix is an invariant w.r.t. the equivalence relation given by similarity. In the classical situation of invariant theory, the equivalence classes are given by the orbits of a group action. In fact, one considers the following setting: G is a linear algebraic group over an algebraically closed field K, and V is a finite-dimensional K-vector space with a linear G-action, given by a morphism G × V → V . In other words, we assume that the action can be described by polynomial functions. A natural extension is to substitute V by an affine K-variety X, which is then called a G-variety. The invariant ring