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

Abstract

Let $F$ be any field of characteristic $p$. It is well-known that there are exactly $p$ inequivalent indecomposable representations $V_1,V_2,...,V_p$ of $C_p$ defined over $F$. Thus if $V$ is any finite dimensional $C_p$-representation there are non-negative integers $0\leq n_1,n_2,..., n_k \leq p-1$ 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 $\SL_2(\C)$ 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 $F[ \oplus_{i=1}^k V_{n_i+1}]^{C_p}$ to the computation of the classical ring of invariants (or covariants) $\C[R_1 \oplus (\oplus_{i=1}^k R_{n_i})]^{\SL_2(\C)}$. This shows that the problem of computing modular $C_p$ invariants is equivalent to the problem of computing classical $\SL_2(\C)$ 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 $F[m V_2]^{C_p}$, $F[m V_3]^{C_p}$ and $F[m V_4]^{C_p}$for all $m \in \N$. This is the first computation of the latter two families of rings of invariants.