Unit groups of representation rings and their ghost rings as inflation functors

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.