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.
Let G be a finite group, p a prime, and (K, O, F ) a p-modular system. We prove that the trivial source ring of OG is isomorphic to the ring of coherent G-stable tuples (χ P ), where χ P is a virtual character of K[N G (P )/P ], P runs through all p-subgroups of G, and the coherence condition is the equality of certain character values. We use this result to describe the group of orthogonal units of the trivial source ring as the product of the unit group of the Burnside ring of the fusion system of G with the group of coherent G-stable tuples (ϕ P ) of homomorphisms N G (P )/P → F × . The orthogonal unit group of the trivial source ring of OG is of interest, since it embeds into the group of p-permutation autoequivalences of OG.
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