A General Theory of Canonical Induction Formulae

AbstractSo far there exist three independent constructions of two different canonical versions of Brauer's induction theorem for complex characters due to V. Snaith, P. Symonds, and the author. “Canonical” in this context means functorial with respect to restrictions along group homomorphisms. In this article we axiomatize the situation in which the above canonical induction formulae are constructed. Mackey functors and related structures arise in this way naturally as a convenient language. This approach allows us to construct canonical induction formulae for arbitrary Mackey functors. In particular we obtain canonical induction formulae for the Brauer character ring, the group of projective characters, the ring of trivial source modules, and the ring of linear source modules. In most cases, it is not difficult to construct such formulae over the rational numbers. A much more subtle question is whether the constructed formula comes from a canonical induction formula defined over the integers. We give a sufficient condition in the general framework of Mackey functors for a canonical induction formula to be integral. As an application we show how canonical induction formulae allow the construction of functorial maps on representation rings in terms of functorial maps on subrings, as, for example, the span of linear characters in the case of the canonical Brauer induction formula. This will be used in a subsequent article in the case of Adams operations and Chern classes