Invariants of Finite Groups and Involutive Division

The invariant ring of a finite matrix group is known to be well behaved for reflection groups and messy in general. Involutive division is a newly discovered tool in commutative algebra and in this note it is applied to the problem of finding a presentation of the ring of invariants of a finite matrix group. The first author has implemented the Janet-algorithm in MAPLE following [GeB 98a] and [GeB 98b], more precisely Gerdt’s involutive algorithm for Janet’s (involutive) division. The results of this are collected in two MAPLE-packages called INVOLUTIVE and JANET, the first dealing with polynomials and the second with linear partial differential equations. Both of these packages have a collection of other routines serving various purposes. There is also a loose connection with the MAPLE-package JETS by Mohammed Barakat, which deals with symmetries of differential equations, conservation laws etc.. Here we report on our experience with applying the package INVOLUTIVE to questions of invariant theory of finite groups. We outline an algorithm constructing a presentation of the ring of invariants of a finite complex matrix group and representing each invariant in a unique way as an expression in the generators. We also report on the limits with the present MAPLE implementation. As far as the invariant theory of finite groups proper is concerned, there is a MAPLE-package available to perform the tasks discussed here, cf. [Kern 98] or [Kern 99], based on Groebner basis techniques and even a very effective implementation in MAGMA.