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Open Access Publications from the University of California

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

  • Author(s): Carman, William Robert
  • Advisor(s): Boltje, Robert
  • et al.

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 unit group of the ghost ring of the Burnside ring defines a biset functor, and becomes a useful tool in studying the Burnside ring unit group functor itself. We are interested in studying the unit group of another representation ring: the trivial source ring of a finite group. Here we show how the unit group of the trivial source ring and its associated ghost ring define biset 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 biset functors, which can be completely determined in the case of the character ring and Brauer character ring.

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