Central idempotents of the bifree and left-free double Burnside ring
type: Item
instance of: scholarly article
title: Central idempotents of the bifree and left-free double Burnside ring
zbMATH Open document ID: 1321.19002
author: Robert Boltje
author: Burkhard Külshammer
published in: Israel Journal of Mathematics
publication date: 2014-10-24T00:00:00Z
full work available at URL: https://arxiv.org/abs/1208.2940
review text: Let \(B^{\triangle}(G, G) \subseteq B^{\triangleleft}(G, G) \subseteq B(G, G)\) be the subrings of the double Burnside ring, the first arises from bifree \((G, G)\)-bisets and the second from left-free \((G, G)\)-bisets. For \(G\) a finite group, these classes of bisets are related to globally defined Mackey functors. The bifree subring of a \(p\)-group is related to fusion systems on the group and the left-free subring is related to the stable homotopy classes of self-maps of the \(p\)-completion of the classifying space. Let \(\widehat{\Sigma}_G\) be a set of representatives of the isomorphism classes of \(G\). In the paper it is proved that the primitive central idempotents of \(B^{\triangle}(G, G)\) are parametrized by elements \(U\in \widehat{\Sigma}_G\) that are perfect. Also, it is proved that the center of \(B^{\triangleleft}(G, G)\) is connected i.e, the only central idempotents are \(0\) and \(1\). The central idempotents of \(\mathbb{Q}B^{\triangle}(G, G)\) are parametrized by pairs \((U, {\chi}) \in \mathcal{E}_G\) with \(U\) as before and \(\chi\) certain characters in \(\text{Irr}_\mathbb{Q}(\text{Out}(U))\). The primitive central idempotents of \(B^{\triangleleft}(G, G)\) are contained in \(\mathbb{Q}B^{\triangle}(G, G)\) and they are the sum of the idempotents in \(\mathbb{Q}B^{\triangle}(G, G)\) over a subset of \(\mathcal{E}_G\). The results actually are given not just for \(\mathbb{Q}\) but for certain integral domains and their fields of fractions. The authors give explicit calculations for cyclic groups and elementary abelian \(p\)-groups.
reviewed by: Stratos Prassidis
Mathematics Subject Classification ID: 19A22
Mathematics Subject Classification ID: 20C15
zbMATH DE Number: 6361331
zbMATH Keywords: double Burnside ring
zbMATH Keywords: bifree bisets
zbMATH Keywords: left-free bisets
MaRDI profile type: Publication
OpenAlex ID: W2067337187
arXiv ID: 1208.2940
cites work: Q3114249
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cites work: On Schur's \(Q\)-functions and the primitive idempotents of a commutative Hecke algebra
DOI: 10.1007/S11856-014-1068-1
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description: scientific article; zbMATH DE number 6361331