2018/06/28 by Elmer, Jonathan · 1 citation
#13A50 #Commutative Algebra (math.AC) #FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.1806.11024
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^*) ⊗ 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.