A general theory of canonical induction formulae
type: Item
instance of: scholarly article
title: A general theory of canonical induction formulae
zbMATH Open document ID: 0913.20001
author: Robert Boltje
published in: Journal of Algebra
publication date: 1999-04-11T00:00:00Z
review text: 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…
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 19A22
zbMATH DE Number: 1214153
zbMATH Keywords: canonical induction formulae
zbMATH Keywords: Mackey functors
zbMATH Keywords: character rings of finite groups
zbMATH Keywords: Burnside rings
zbMATH Keywords: representation rings
zbMATH Keywords: Brauer induction
reviewed by: Andrei Marcus
MaRDI profile type: Publication
full work available at URL: jabr.1998.7419
OpenAlex ID: W2022849953
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DOI: 10.1006/JABR.1998.7419
description: scientific article; zbMATH DE number 1214153