Modular covariants of cyclic groups of order p

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 V_n (n = 1, . . . , p) and V_n has dimension n. For any n, we show that k[V_2,V_n]^G is a free k[V_2]^G- module (recovering a result of Broer and Chuai) and we give an explicit set of covariants generating k[V_2,V_n]^G freely over k[V_2]^G . For any n, we show that k[V_3,V_n]^G is a Cohen-Macaulay k[V_3]^G -module (again recovering a result of Broer and Chuai) and we give an explicit set of covariants which generate k[V 3 , V n ] G freely over a homogeneous system of parameters for k[V_3]^G . We also use our results to compute a minimal generating set for the transfer ideal of k[V_3]^G over a homogeneous system of parameters.

© 2022 The Author. Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).