On tensor products of indecomposable representations of the cyclic group of prime order \(p\) over the ring \(\mathbb{Z}_p[[x]]\)

Abstract

Let \(C_p\) be a cyclic group of prime order \(p\), let \(R=\mathbb{Z}_p[[x]]\) be the ring of formal power series in one indeterminate \(x\) with coefficients in the ring \(\mathbb{Z}_p\) of \(p\)-adic integers, and let \(RC_p\) be the corresponding group ring. We determine the decompositions into indecomposable summands of the tensor products over \(R\) of all indecomposable \(RC_p\)-modules that are free as \(R\)-modules. The resulting formulas cover tensor products involving the regular module, the cyclotomic module, the families \(\Gamma_i\), and their contragredient modules, including tensor squares and mixed products. For odd \(p\), the proof is based on exact sequences, suitable reductions of modules, and decompositions of symmetric and exterior squares. The case \(p=2\) is covered by earlier direct computations.