Computing Modular Invariants of p-groups

Let V be a finite dimensional representation of a p-group, G, over a field, k, of characteristic p. We show that there exists a choice of basis and monomial order for which the ring of invariants, k[V ]G, has a finite SAGBI basis. We describe two algorithms for constructing a generating set for k[V ]G. We use these methods to analyse k[2V3]3 where U3 is the p-Sylow subgroup of GL3(Fp) and 2V3 is the sum of two copies of the canonical representation. We give a generating set for k[2V3]3 for p = 3 and prove that the invariants fail to be Cohen–Macaulay for p > 2. We also give a minimal generating set for k[mV2] were V2 is the two-dimensional indecomposable representation of the cyclic group Z/p. c © 2002 Elsevier Science Ltd. All rights reserved.