Periodicity of Adams operations on the Green ring of a finite group

Abstract

The Adams operations $ψ_Λ^n$ and $ψ_S^n$ on the Green ring of a group $G$ over a field $K$ provide a framework for the study of the exterior powers and symmetric powers of $KG$-modules. When $G$ is finite and $K$ has prime characteristic $p$ we show that $ψ_Λ^n$ and $ψ_S^n$ are periodic in $n$ if and only if the Sylow $p$-subgroups of $G$ are cyclic. In the case where $G$ is a cyclic $p$-group we find the minimum periods and use recent work of Symonds to express $ψ_S^n$ in terms of $ψ_Λ^n$.