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

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

2009-12-15

arxiv.org/abs/0912.2933v1

Mathematics

Representation Theory

Scientific paper

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 $\psi_\Lambda^n$.

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