Graded character rings of finite groups

Abstract

Let $G$ be a finite group. The ring $R_{\mathbb{K}}(G)$ of virtual characters of $G$ over the field $\mathbb{K}$ is a $\lambda$-ring; as such, it is equipped with the so-called $\Gamma$-filtration, first defined by Grothendieck. We explore the properties of the associated graded ring $R_{\mathbb{K}}^{\ast }(G)$, and present a set of tools to compute it through detailed examples. In particular, we use the functoriality of $R_{\mathbb{K}}^{\ast }(-)$, and the topological properties of the $\Gamma$-filtration, to explicitly determine the graded character ring over the complex numbers of every group of order at most $8$, as well as that of dihedral groups of order $2p$ for $p$ prime.