Modular Covariants of Cyclic Groups of Order p

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 $k[V,W{]}^G=(S(V^{\ast })\otimes W{)}^G$. 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.

18 pages