Modular Covariants of Cyclic Groups of Order p

TL;DR

This paper investigates the structure of covariant modules for cyclic groups of prime order over fields of characteristic p, providing explicit generators and demonstrating freeness and Cohen-Macaulay properties in specific cases.

Contribution

It extends previous results by explicitly constructing generators and establishing freeness and Cohen-Macaulay properties for covariant modules in new cases of indecomposable modules.

Findings

$k[V,W]^G$ is a free $k[V]^G$-module for certain indecomposable modules of dimension 2.

$k[V,W]^G$ is Cohen-Macaulay over $k[V]^G$ for indecomposable modules of dimension 3 with $W$ up to dimension 5.

Explicit generators for covariant modules are provided in the studied cases.

Abstract

Let $G$ be a cyclic group of order $p$, let $k$ be a field of characteristic $p$, and let $V, W$ be $kG$-modules. We study the modules of covariants $k[V,W]^G = (S(V^*) \otimes W)^G$. For $V$ indecomposable with dimension 2, and $W$ an arbitrary indecomposable module, we show $k[V,W]^G$ is a free $k[V]^G$-module (recovering a result of Broer and Chuai) and we give an explicit set of covariants generating $k[V,W]^G$ freely over $k[V]^G$. For $V$ indecomposable with dimension 3 and $W$ an indecomposable module with dimension at most 5, we show that $k[V,W]^G$ is a Cohen-Macaulay $k[V]^G$-module (again recovering a result of Broer and Chuai) and we give an explicit set of covariants which generate $k[V,W]^G$ freely over a homogeneous system of parameters for $k[V]^G$. We conjecture that a similar set of covariants generates $k[V,W]^G$ freely over a homogeneous system of parameters for $k[V]^G$ when $W$ has arbitrary dimension.