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dc.contributor.authorBoltje, Robert
dc.contributor.authorDanz, Susanne
dc.date.accessioned2012-07-03T12:00:00Z
dc.date.accessioned2016-10-05T14:14:23Z
dc.date.available2012-07-03T12:00:00Z
dc.date.available2016-10-05T14:14:23Z
dc.date.issued2012-07-03
dc.identifier.urihttp://publications.mfo.de/handle/mfo/1212
dc.descriptionResearch in Pairs 2012en_US
dc.description.abstractFor a finite group $G$, we introduce a multiplication on the $\mathbb{Q}$-vector space with basis $\mathscr{S}_{G\times G}$, the set of subgroups of ${G \times G}$. The resulting $\mathbb{Q}$-algebra $\tilde{A}$ can be considered as a ghost algebra for the double Burnside ring $B(G,G)$ in the sense that the mark homomorphism from $B(G,G)$ to $\tilde{A}$ is a ring homomorphism. Our approach interprets $\mathbb{Q}B(G,G)$ as an algebra $eAe$, where $A$ is a twisted monoid algebra and $e$ is an idempotent in $A$. The monoid underlying the algebra $A$ is again equal to $\mathscr{S}_{G\times G}$ with multiplication given by composition of relations (when a subgroup of $G \times G$ is interpreted as a relation between $G$ and $G$). The algebras $A$ and $\tilde{A}$ are isomorphic via Möbius inversion in the poset $\mathscr{S}_{G\times G}$. As an application we improve results by Bouc on the parametrization of simple modules of $\mathbb{Q}B(G,G)$ and also of simple biset functors, by using results by Linckelmann and Stolorz on the parametrization of simple modules of finite category algebras. Finally, in the case where G is a cyclic group of order n, we give an explicit isomorphism between $\mathbb{Q}B(G,G)$ and a direct product of matrix rings over group algebras of the automorphism groups of cyclic groups of order $k$, where $k$ divides $n$.en_US
dc.language.isoenen_US
dc.publisherMathematisches Forschungsinstitut Oberwolfachen_US
dc.relation.ispartofseriesOberwolfach Preprints;2012,09
dc.subjectBurnside ringen_US
dc.subjectDouble Burnside ringen_US
dc.subjectMark homomorphismen_US
dc.subjectGhost ringen_US
dc.subjectSchur functoren_US
dc.subjectBiseten_US
dc.subjectBiset functoren_US
dc.subjectTwisted category algebraen_US
dc.titleGhost Algebras of Double Burnside Algebras via Schur Functorsen_US
dc.typePreprinten_US
dc.identifier.doi10.14760/OWP-2012-09
local.scientificprogramResearch in Pairs 2012
local.series.idOWP-2012-09
local.subject.msc19
local.subject.msc20


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