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

TL;DR

This paper provides a new proof for the minimal generating set of vector invariants for a 2D indecomposable representation of a cyclic group of prime order, also establishing a SAGBI basis and explicit module decomposition.

Contribution

It introduces a new proof for the minimal generating set, proves it forms a SAGBI basis, and explicitly decomposes the module, extending understanding of invariants in modular representation theory.

Findings

Established a minimal generating set for the invariants.

Proved the generating set is a SAGBI basis.

Explicitly decomposed the module into indecomposables.

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 ${\bf{F}}[m V_2]^{SL_2({\bf F}_p)}$.