Minimal generating sets of non-modular invariant rings of finite groups

Abstract

It is a classical problem to compute a minimal set of invariant polynomials generating the invariant ring of a finite group as a sub-algebra. We present here a new algorithmic solution in the non-modular case.Our algorithm only involves very basic operations and is based on well-known ideas. In contrast to the algorithm of Kemper and Steel, it does not rely on the computation of primary and (irreducible) secondary invariants. In contrast to the algorithm of Thiéry, it is not restricted to permutation representations.With the first implementation of our algorithm in Singular, we obtained minimal generating sets for the natural permutation action of the cyclic groups of order up to 12 in characteristic 0 and of order up to 15 for finite fields. This was far out of reach for implementations of previously described algorithms. By now our algorithm has also been implemented in Magma.As a by-product, we obtain a new algorithm for the computation of irreducible secondary invariants that, in contrast to previously studied algorithms, does not involve a computation of all reducible secondary invariants.