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

Abstract

In this paper, we study the vector invariants, F[mV_2]^(C_p), of the 2-dimensional indecomposable representation V_2 of the cylic group, C_p, of order p over a field F of characteristic p. This ring of invariants was first studied by David Richman 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. Later, Shank and Wehlau 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 V_2]^(C_p). In fact, our proof does much more. We show that our minimal generating set is also a SAGBI basis. Further, our techniques also serve to give an explicit decomposition of F[m V_2] into a direct sum of indecomposable C_p-modules. Finally, noting that our representation of C_p on V_2 is as the p-Sylow subgroup of SL_2(F_p), we are able to determine a generating set for the ring of invariants of F[m V_2]^(SL_2(F_p))