The Module Structure of a Group Action on a Polynomial Ring: a Finiteness Theorem

We consider a polynomial ring S in n variables over a finite field k of characteristic p and an action of a finite group G on S by homogeneous linear substitutions. This is equivalent to taking the symmetric powers of an n-dimensional kG-module. We want to understand S as a kG-module in a manner as explicit as possible. The ideal solution would be to give a decomposition into indecomposable summands. We are primarily interested in the modular case, when p divides the order of G, so the problem is much harder than that of determining the composition factors. The case of two variables was studied by Glover [13] and Alperin and Kovacs [2] and the case of three variables by the authors in [11]. This paper generalizes the results of [11] to any number of variables, and we prove a strong finiteness property as a consequence.