Invariants for the Modular Cyclic Group of Prime Order via Classical Invariant Theory

TL;DR

This paper proves a conjecture linking the computation of modular cyclic group invariants to classical SL(2,C) invariants, enabling new calculations of invariant rings for various representations.

Contribution

It reduces the problem of computing modular cyclic group invariants to classical invariant theory, allowing explicit calculations for the first time.

Findings

Computed invariant rings for multiple representations of C_p.

Established equivalence between modular invariants and classical SL_2 invariants.

Provided generators for vector invariants F[m V_2]^{C_p}, F[m V_3]^{C_p}, and F[m V_4]^{C_p}.

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.