Modular covariants of cyclic groups of order p

Abstract

Let G be a cyclic group of order p and let V,W be kG-modules. We study the modules of covariants k[V,W]G=(S(V⁎)⊗W)G. Recall that G has exactly p inequivalent indecomposable kG-modules, denoted Vn (n=1,…,p) and Vn has dimension n. For any n, we show that k[V2,Vn]G is a free k[V2]G-module (recovering a result of Broer and Chuai [1]) and we give an explicit set of covariants generating k[V2,Vn]G freely over k[V2]G. For any n, we show that k[V3,Vn]G is a Cohen-Macaulay k[V3]G-module (again recovering a result of Broer and Chuai) and we give an explicit set of covariants which generate k[V3,Vn]G freely over a homogeneous system of parameters for k[V3]G. We also use our results to compute a minimal generating set for the transfer ideal of k[V3]G over a homogeneous system of parameters.