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

TL;DR

This paper investigates the periodicity of Adams operations on the Green ring of a finite group over a field of prime characteristic, linking it to the structure of Sylow p-subgroups and providing explicit period calculations.

Contribution

It establishes a precise criterion for the periodicity of Adams operations based on Sylow p-subgroup structure and derives explicit periods for cyclic p-groups.

Findings

Adams operations are periodic if and only if Sylow p-subgroups are cyclic.

Explicit minimum periods are determined for cyclic p-groups.

Expresses symmetric powers in terms of exterior powers using recent results.

Abstract

The Adams operations $\psi_\Lambda^n$ and $\psi_S^n$ 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 $\psi_\Lambda^n$ and $\psi_S^n$ 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 $\psi_S^n$ in terms of $\psi_\Lambda^n$.