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 analysek [2V3 ]U3whereU3is the p -Sylow subgroup ofGL3 (Fp) and 2 V3is the sum of two copies of the canonical representation. We give a generating set for k [2 V3]U3forp = 3 and prove that the invariants fail to be Cohen–Macaulay forp > 2. We also give a minimal generating set for k [mV2 ]Z / pwereV2is the two-dimensional indecomposable representation of the cyclic group Z / p.