Symmetric powers of the p+1-dimensional indecomposable module of a cyclic p-group and the λ-structure of its Green ring

Let K be a field of order p and let G be a cyclic group of order pk (k⩾2). We explicitly decompose the symmetric powers of the indecomposable KG-module of dimension p+1 into indecomposable KG-modules. Using this result, for every odd prime p, we give a negative answer to the conjecture posed by Kouwenhoven [F.M. Kouwenhoven, The λ-structure of the Green rings of cyclic p-groups, Proc. Sympos. Pure Math. 47 (1987) 451–466].