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

Abstract

It is a classical problem to compute a minimal set of invariant polynomial generating the invariant ring of a finite group as an algebra. We present here an algorithm for the computation of minimal generating sets in the non-modular case. Apart from very few explicit computations of Groebner bases, the algorithm only involves very basic operations, and is thus rather fast. As a test bed for comparative benchmarks, we use transitive permutation groups on 7 and 8 variables. In most examples, our algorithm implemented in Singular works much faster than the one used in Magma, namely by factors between 50 and 1000. We also compute some further examples on more than 8 variables, including a minimal generating set for the natural action of the cyclic group of order 11 in characteristic 0 and of order 15 in characteristic 2. We also apply our algorithm to the computation of irreducible secondary invariants.