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 ]^U₃where U₃is the p -Sylow subgroup ofGL₃ (F_p) 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 / pwere V₂is the two-dimensional indecomposable representation of the cyclic group Z / p.