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

Abstract

In this paper, we study the vector invariants, \({\bf{F}}[m V_2]^{C_p}\), of the 2-dimensional indecomposable representation \(V_2\) of the cylic group, \(C_p\), of order \(p\) over a field \({\bf{F}}\) of characteristic \(p\). This ring of invariants was first studied by David Richman \cite{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 in \cite{campbell-hughes}. Later, Shank and Wehlau in \cite{cmipg} 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 \({\bf{F}}[m V_2]^{C_p}\). In fact, our proof does much more. We show that our minimal generating set is also a SAGBI basis for \({\bf{F}}[m V_2]^{C_p}\). Further, our techniques also serve to give an explicit decomposition of \({\bf{F}}[m V_2]\) 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 \(SL_2({\bf F}_p)\), we are able to determine a generating set for the ring of invariants of \({\bf{F}}[m V_2]^{SL_2({\bf F}_p)}\).