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 fork[V]G. We use these methods to analyse k[2V(3)](U3) where U-3 is the p-Sylow subgroup of GL(3)(F-p) and 2V(3) is the sum of two copies of the canonical representation. We give a generating set for k[2V(3)](U3) 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[mV(2)](Z/p) were V-2 is the two-dimensional indecomposable representation of the cyclic group Z/p. (C) 2002 Elsevier Science Ltd.. All rights reserved.