Invariants for the modular cyclic group of prime order via classical invariant theory

Abstract

Let $\mathbb{F}$ be any field of characteristic $p$. It is well-known that there are exactly $p$ inequivalent indecomposable representations $V_1,V_2,\dots ,V_p$ of $C_p$ defined over $\mathbb{F}$. Thus if $V$ is any finite dimensional $C_p$-representation there are non-negative integers $0\le n_1,n_2,\dots ,n_k\le p-1$ such that $V\cong {\oplus }_{i=1}^k{V}_{n_i+1}$. It is also well-known there is a unique (up to equivalence) $d+1$ dimensional irreducible complex representation of SL${}_2(\mathbb{C})$ given by its action on the space $R_d$ of $d$ forms. Here we prove a conjecture, made by R. J. Shank, which reduces the computation of the ring of $C_p$-invariants $\mathbb{F}[{\oplus }_{i=1}^k{V}_{n_i+1}{]}^{C_p}$ to the computation of the classical ring of invariants (or covariants) $\mathbb{C}[R_1\oplus ({\oplus }_{i=1}^k{R}_{n_i}){]}^{{\mathrm{SL}}_2(\mathbb{C})}$. This shows that the problem of computing modular $C_p$ invariants is equivalent to the problem of computing classical SL${}_2(\mathbb{C})$ invariants. This allows us to compute for the first time the ring of invariants for many representations of $C_p$. In particular, we easily obtain from this generators for the rings of vector invariants $\mathbb{F}[m{V}_2{]}^{C_p}$, $\mathbb{F}[m{V}_3{]}^{C_p}$ and $\mathbb{F}[m{V}_4{]}^{C_p}$for all $m\in \mathbb{N}$. This is the first computation of the latter two families of rings of invariants.