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 $R^*_\mathbb{K}(G)$, and present a set of tools to compute it through detailed examples. In particular, we use the functoriality of $R^*_\mathbb{K}(-)$, 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.

中文翻译：

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有限群的分级字符环

令 $G$ 是一个有限群。域$\mathbb{K}$上$G$的虚拟字符的环$R_\mathbb{K}(G)$是$\lambda$-环；因此，它配备了所谓的 $\Gamma$-过滤，首先由 Grothendieck 定义。我们探索了相关分级环 $R^*_\mathbb{K}(G)$ 的属性，并通过详细示例提供了一组计算它的工具。特别地，我们使用 $R^*_\mathbb{K}(-)$ 的函子性和 $\Gamma$-过滤的拓扑性质，来明确地确定每组复数上的分级字符环最多 $8$ 的顺序，以及 $p$prime 的顺序为 $2p$ 的二面体组。