A general theory of canonical induction formulae
The aim of this paper is to introduce a general setting in which canonical induction formulae for representation rings can be produced. Until now three independent constructions of two different canonical (that is, functorial with respect to group homomorphisms) versions of Brauer's induction theorem were given by \textit{V. Snaith} [Invent. Math. 94, No. 3, 455-478 (1988; Zbl 0704.20009)], \textit{P. Symonds} [Comment. Math. Helv. 66, No. 2, 169-184 (1991; Zbl 0797.20008)], and \textit{R. Boltje} [Astérisque 181-182, 31-59 (1990; Zbl 0718.20005)]. The author gives here the definitions, constructions and results of this general theory, and the paper is also aimed to be a reference for future examples and applications. The paper is divided into 11 sections. In Section 1 the language of Mackey functors on a finite group \(G\) over a commutative base ring \(k\) is introduced, this being suited to formulate subsequent results. In Section 2 two basic functors \(-_+\) and \(-^+\) between three categories associated to \(G\) are defined. Let \(M\) be a Mackey functor, \(A\) a restriction subfunctor of \(M\), \(b\colon A_+\to M\) the induction morphism, where \(A_+(G)\) is a quotient of \(\bigoplus_{H\leq G}A(H)\) and \(b_G\colon A_+(G)\to M(G)\) is induced by the induction maps. With these notations, in Section 3 a canonical induction formula is defined as a morphism of restriction functors \(a\colon M\to A_+\) which splits \(b\). As an application, a canonical induction formula is used in Section 4 to extend morphisms on \(A\) to morphisms on \(B\). For suitable base rings, all morphisms of restriction functors \(a\) are parametrized in Section 5 by the set of morphisms \(p\colon M\to A\) of conjugation functors. This proves to be particularly useful in the case considered in Section 6, when the order of \(G\) is invertible in \(k\). In Section 7 the change of base rings, especially passing from \(\mathbb{Q}\) to \(\mathbb{Z}\), is investigated. A standard situation (with an additional condition on \(M\) and \(A\)) is studied in Section 8, and a method of computing a canonical induction formula in this situation is given in Section 11. A general tool for proving integrality results is provided in Section 9. In Section 10 the setting is generalized even more by considering \(M\) and \(A\) to be defined on all finite groups instead of the subgroups of a given group.
scientific article; zbMATH DE number 1214153
scientific article