Adjoint functors and an extension of the Green correspondence for group representations
type: Item
instance of: scholarly article
title: Adjoint functors and an extension of the Green correspondence for group representations
zbMATH Open document ID: 0807.20009
author: Maurice Auslander
author: Mark Kleiner
published in: Advances in Mathematics
publication date: 1995-03-07T00:00:00Z
review text: The results of this paper are inspired by the observation that the proofs of the Green correspondence and the Burry-Carlson-Puig theorem rely on the Mackey decomposition and on the fact that induction and restriction are adjoint functors. The main results here are then a Green correspondence and Burry-Carlson-Puig theorem for adjoint functors. The general setting is as follows: consider additive categories \(\mathcal D\), \(\mathcal H\), \(\mathcal G\) and functors \(S : {\mathcal H} \to {\mathcal G}\), \(T : {\mathcal G} \to {\mathcal H}\), \(S' : {\mathcal D} \to {\mathcal H}\), \(T' : {\mathcal H} \to {\mathcal D}\), such that \(S\) (\(S'\)) is a left adjoint of \(T\) (\(T'\)), and \(TS = I_{\mathcal H} \oplus U\), and let finally \(\mathcal Y\) be a suitable subcategory of \(\mathcal H\). The Green correspondence is an equivalence between certain quotient categories defined with the aid of the above data. For the Burry-Carlson-Puig type theorem, \(S\) is assumed to be both a left and a right adjoint of \(T\), and \(\mathcal H\), \(\mathcal G\) to be Krull- Schmidt categories. Two statements of uniqueness concerning the Green correspondence are proved, one of them being new even for group representations. The applications to group algebras are obtained by taking the groups \(D \leq H \leq G\), the categories \({\mathcal D} =\text{mod-}RD\), \({\mathcal H} =\text{mod-}RH\), \({\mathcal G} =\text{mod-}RG\) (where \(R\) is a noetherian commutative ring), the restriction functors \(S\), \(S'\) and the induction functors \(T\), \(T'\), and also two different choices for \(\mathcal Y\) are considered. It is to be mentioned that the general results not only throw a new light on known facts, but also provide new results in the particular cases.
reviewed by: Andrei Marcus
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 18A40
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 18E05
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 16D90
zbMATH DE Number: 611205
zbMATH Keywords: modules over group algebras
zbMATH Keywords: Green correspondence
zbMATH Keywords: Burry-Carlson-Puig theorem
zbMATH Keywords: Mackey decomposition
zbMATH Keywords: induction
zbMATH Keywords: restriction
zbMATH Keywords: adjoint functors
zbMATH Keywords: quotient categories
zbMATH Keywords: Krull-Schmidt categories
zbMATH Keywords: uniqueness
zbMATH Keywords: group representations
zbMATH Keywords: group algebras
zbMATH Keywords: restriction functors
zbMATH Keywords: induction functors
MaRDI profile type: Publication
full work available at URL: aima.1994.1031
OpenAlex ID: W1984498282
Wikidata QID: Q105542153
DOI: 10.1006/AIMA.1994.1031
description: scientific article; zbMATH DE number 611205