Calculating Invariant Rings of Finite Groups over Arbitrary Fields

If G is a finite linear group over a field K such that char(K) | 6 |G|, there are various effective methods to calculate the invariant ring I of G, i.e., to find a finite system of generators of I as an algebra over K (see Sturmfels, 1993, Section 2; McShane, 1992; Kemper, 1993, 1994). These methods make use of the Reynolds operator, Molien’s formula, the Cohen–Macaulay property of invariant rings, and of Noether’s degree bound in the case char(K) > |G| (Noether, 1916). If, on the other hand, |G| is a multiple of char(K) (which we shall call the modular case), none of these techniques are available since they all involve divisions by |G|. In fact, G is not a linearly reductive group in this case. Nevertheless, the invariant ring is finitely generated as a K-algebra (Noether, 1926). If G is a permutation group there is a very nice algorithm by Göbel (1995) which calculates generators over any commutative ground ring and which implies a degree bound for the generators. But for the case of finite linear groups there is no algorithm available at the moment to construct a system of generators, and modular invariant rings are calculated by ad hoc methods (see Benson, 1993, Chapter 8; Wilkerson, 1983; Adem and Milgram, 1994, Chapter III). The purpose of this paper is to fill this gap. With the knowledge of generators for the invariant ring, it is quite easy to calculate its depth and its Poincaré series (see Section 4.1). It is even easier to check the Cohen–Macaulay property. So the algorithm presented here could be useful to gain some experience and to test hypotheses. The first section of this paper is concerned with the calculation of primary invariants, which serve as a kind of first approximation to the invariant ring. A new algorithm to