Unit Groups of Representation Rings and their Ghost Rings as Inflation Functors

TL;DR

This paper explores the structure of unit groups of various representation rings of finite groups, showing they form inflation functors and analyzing their torsion subgroups, extending the biset functor framework.

Contribution

It introduces the study of unit groups of the trivial source ring and its ghost ring as inflation functors, connecting them with Burnside, character, and Brauer character rings.

Findings

Unit groups of trivial source and ghost rings define inflation functors.

Restriction to torsion subgroups yields inflation functors, fully determined for character and Brauer rings.

The framework links different representation rings through their unit groups and inflation functors.

Abstract

The theory of biset functors developed by Serge Bouc has been instrumental in the study of the unit group of the Burnside ring of a finite group, in particular for the case of p-groups. The ghost ring of the Burnside ring defines an inflation functor, and becomes a useful tool in studying the Burnside ring functor itself. We are interested in studying the unit group of another representation ring: the trivial source ring of a finite group. In this article, we show how the unit groups of the trivial source ring and its associated ghost ring define inflation functors. Since the trivial source ring is often seen as connecting the Burnside ring to the character ring and Brauer character ring of a finite group, we study all these representation rings at the same time. We point out that restricting all of these representation rings' unit groups to their torsion subgroups also give inflation functors, which we can completely determine in the case of the character ring and Brauer character ring.