Vector Invariants for the Two Dimensional Modular Representation of a Cyclic Group of Prime Order

In this paper, we study the vector invariants, F[m V2] Cp , of the 2-dimensional indecomposable representation V2 of the cylic group, Cp, of order p over a field F of characteristic p. This ring of invariants was first studied by David Richman [18] who showed that this 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 in [2]. Later, Shank and Wehlau in [21] 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 F[m V2] Cp . In fact, our proof does much more. We show that our minimal generating set is also a SAGBI basis for F[m V2] Cp . Further, our techniques also serve to give an explicit decomposition of F[m V2] into a direct sum of indecomposable Cpmodules. Finally, noting that our representation of Cp on V2 is as the p-Sylow subgroup of SL2(Fp), we are able to determine a generating set for the ring of invariants of F[m V2] 2p.