The slice Burnside ring and the section Burnside ring of a finite group
type: Item
instance of: scholarly article
title: The slice Burnside ring and the section Burnside ring of a finite group
zbMATH Open document ID: 1252.19001
DOI: 10.1112/S0010437X11007500
author: Serge Bouc
published in: Compositio Mathematica
publication date: 2012-06-29T00:00:00Z
full work available at URL: https://arxiv.org/abs/1101.3138
Mathematics Subject Classification ID: 19A22
Mathematics Subject Classification ID: 18F30
zbMATH DE Number: 6051029
zbMATH Keywords: Burnside ring
zbMATH Keywords: ghost map
zbMATH Keywords: Green biset functor
MaRDI profile type: Publication
arXiv ID: 1101.3138
cites work: A characterisation of solvable groups
cites work: Ring of subquotients of a finite group. I: Linearization
cites work: On the unit groups of Burnside rings
cites work: The functor of units of Burnside rings for \(p\)-groups
cites work: Construction of functors between categories of \(G\)-sets
cites work: Biset functors for finite groups
OpenAlex ID: W3100093491
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review text: The Burnside ring \(B(G)\) of a finite group \(G\) is the Grothendieck ring of the category of finite \(G\)-sets. In the paper under review, the author introduces two variants of this construction, the slice Burnside ring \(\Xi(G)\) and the section Burnside ring \(\Gamma(G)\). Here \(\Xi(G)\) is the Grothendieck ring of the category of all morphisms between finite \(G\)-sets, and \(\Gamma(G)\) is the Grothendieck ring of the category of all Galois morphisms between finite \(G\)-sets where a morphism of finite \(G\)-sets \(f:X \longrightarrow Y\) is called a Galois morphism if \(G_x = G_{x'}\) for all \(x,x' \in G\) such that \(f(x) = f(x')\).NEWLINENEWLINEThe author develops a structure theory for \(\Xi(G)\) and \(\Gamma(G)\) which parallels the corresponding theory for \(B(G)\). Both are commutative rings, and free of finite rank as abelian groups; moreover, \(\Gamma(G)\) is a subring of \(\Xi(G)\). Both \(\Xi(G)\) and \(\Gamma(G)\) can be embedded into ghost rings, and the images of these embeddings can be characterized via certain congruences. Moreover, there are ring homomorphisms \(s_G:\Xi(G) \longrightarrow B(G)\) and \(i_G:B(G) \longrightarrow \Xi(G)\) such that the image of \(i_G\) is contained in \(\Gamma(G)\) and \(s_G \circ i_G = \text{id}_{B(G)}\). Tensoring with \(\mathbb{Q}\) leads to split semisimple \(\mathbb{Q}\)-algebras \(\mathbb{Q} \Xi (G)\) and \(\mathbb{Q} \Gamma (G)\). Explicit formulas for the idempotents of \(\mathbb{Q} \Xi (G)\) and \(\mathbb{Q} \Gamma (G)\) can be given, and the prime spectra of \(\Xi(G)\) and \(\Gamma(G)\) can be determined. The correspondences \(G \longmapsto \Xi(G)\) and \(G \longmapsto \Gamma(G)\) define Green biset functors. The author also deals with the unit groups of \(\Xi(G)\) and \(\Gamma(G)\). They are elementary abelian finite \(2\)-groups. However, in contrast to the unit groups of \(B(G)\), they cannot be endowed with a biset functor structure.
reviewed by: Burkhard Külshammer
description: scientific article; zbMATH DE number 6051029