Modular covariants of cyclic groups of order \(p\) (Q2667907)

Let \(G\) be a cyclic group of prime order \(p\), \(\Bbbk\) be a field of characteristic \(p\), and \(V, W\) be finite dimensional \(\Bbbk G\)-modules, therefore we have a modular representation. In this note, the author provides constructive methods for the structures and generators of the covariants \(\Bbbk[V,W]^G\), when \(V\) is the indecomposable \(\Bbbk G\)-module of dimension 2 or 3. For the notations mentioned here, the reader may refer to any standard text on invariant theory or better to the note under review. The methods used in this note can be listed as; the twisted derivation \(\Delta \), Hilbert series \(H(M,t)\), and the map \(\Theta:K_n\to\Bbbk [V,W]^G\) defined as \(\Theta(f)=\sum_{i=1}^n \Delta^{i-1}(f) w_i\) which provides an isomorphism between \(K_n\) and \(\Bbbk [V,W]^G\), and hence used to describe the module generators. To be more specific, \(\Bbbk [V_2,W]^G\) is freely generated over \(\Bbbk [V]^G\) by \(\{\Theta(1)=w_1, \Theta(x_1), \Theta(x_1^2), \dotsc, \Theta(x_1^{n-1})\}\). And, similarly, \(\Bbbk [V_3,W]^G\) is freely generated over \(A\), where \(A\) is a polynomial ring generated by a homogeneous system of parameters, by \(\{\Theta(M_0), \Theta(M_1), \dotsc, \Theta(M_{n-1}), \Theta(P_0), \Theta(\Delta^{p-n}(P_1)),\dotsc,\Theta(\Delta^{p-n}(P_{n-1}))\}\), where \(M_i, P_j\) are explicitly defined in the article. Finally, the author provides an application of the generators of covariants to find the minimal generators of the transfer ideals \(\mathrm{Tr}^G(\Bbbk [V_2])\) and \(\mathrm{Tr}^G(\Bbbk [V_3])\).