Graded character rings of finite groups

TL;DR

This paper investigates the structure of graded character rings of finite groups, providing tools and explicit computations for groups up to order 8 and certain dihedral groups, enhancing understanding of their algebraic properties.

Contribution

It introduces methods to compute the graded character ring using functoriality and topological properties, with explicit calculations for small and specific groups.

Findings

Explicit description of graded character rings for groups of order up to 8.

Determination of graded character rings for dihedral groups of order 2p.

Development of computational tools for analyzing the $ ext{Gamma}$-filtration}.

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 $R^*_\mathbb{K}(G)$, 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.