Vector invariants for the two-dimensional modular representation of a cyclic group of prime order

In this paper, we study the vector invariants of the 2-dimensional indecomposable representation V2 of the cyclic group, Cp, of order p over a field F of characteristic p, FCp[mV2]. This ring of invariants was first studied by David Richman (1990) [20], who showed that the ring required a generator of degree m(p−1), thus demonstrating that the result of Noether in characteristic 0 (that the ring of invariants of a finite group is always generated in degrees less than or equal to the order of the group) does not extend to the modular case. He also conjectured that a certain set of invariants was a generating set with a proof in the case p=2. This conjecture was proved by Campbell and Hughes (1997) in [3], . Later, Shank and Wehlau (2002) in [24] determined which elements in Richman's generating set were redundant thereby producing a minimal generating set.We give a new proof of the result of Campbell and Hughes, Shank and Wehlau giving a minimal algebra generating set for the ring of invariants FCp[mV2]. In fact, our proof does much more. We show that our minimal generating set is also a SAGBI basis for FCp[mV2]. Further, our results provide a procedure for finding an explicit decomposition of F[mV2] into a direct sum of indecomposable Cp-modules. Finally, noting that our representation of Cp on V2 is as the p-Sylow subgroup of SL2(Fp), we describe a generating set for the ring of invariants F[mV2]SL2(Fp) and show that (p+m−2)(p−1) is an upper bound for the Noether number, for m>2.