Fibred permutation sets and the idempotents and units of monomial Burnside rings
type: Item
instance of: scholarly article
title: Fibred permutation sets and the idempotents and units of monomial Burnside rings
zbMATH Open document ID: 1056.19001
author: Laurence Barker
published in: Journal of Algebra
publication date: 2004-11-26T00:00:00Z
review text: Let \(G\) be a finite group and \(A\) an abelian group. An \(A\)-fibred \(G\)-set is an \(A\)-free \(A\times G\)-set with only finitely many orbits. The category of \(A\)-fibred \(G\)-sets is with products and coproducts, and the corresponding Grothendieck ring \(B(A,G)\) is called the monomial Burnside ring for \(G\) with fibre group \(A\). The author mainly considers the case when the fibre group is a supercyclic group \(C\), that is, isomorphic to a subgroup of \({\mathbb Q}/{\mathbb Z}\). An important part of the paper is devoted to the study of the idempotents of the monomial Burnside algebra \(K\otimes_{\mathbb Z}B(C,G)\), where \(K\) is a field of characteristic zero, by employing a monomial version of the Möbius inversion. The main result of the paper is concerned with the group of units \(B(C,G)^*\) of the monomial Burnside ring. The author constructs a tenduction map, and proves that the functor \(B(C,-)^*\) defined on the category of inclusions of finite groups, and equipped with the tenduction, restriction and conjugation maps, is a Mackey functor over \(\mathbb Z\).
reviewed by: Andrei Marcus
Mathematics Subject Classification ID: 19A22
Mathematics Subject Classification ID: 20C05
zbMATH DE Number: 2119113
zbMATH Keywords: monomial Burnside rings
zbMATH Keywords: group of units
zbMATH Keywords: fibred permutation sets
zbMATH Keywords: tensor induction
zbMATH Keywords: Mackey functors
zbMATH Keywords: units of Burnside rings
zbMATH Keywords: Möbius inversion
zbMATH Keywords: tenduction
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2004.07.022
OpenAlex ID: W2165006486
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DOI: 10.1016/J.JALGEBRA.2004.07.022
description: scientific article; zbMATH DE number 2119113