A ghost algebra of the double Burnside algebra in characteristic zero

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• 作者：Robert Boltje ; Susanne Danz
• 关键词：19A22 ; 20C20
• 刊名：Journal of Pure and Applied Algebra
• 出版年：2013
• 出版时间：April, 2013
• 年：2013
• 卷：217
• 期：4
• 页码：608-635
• 全文大小：451 K

文摘

For a finite group , we introduce a multiplication on the -vector space with basis , the set of subgroups of . The resulting -algebra can be considered as a ghost algebra for the double Burnside ring in the sense that the mark homomorphism from to is a ring homomorphism. Our approach interprets as an algebra , where is a twisted monoid algebra and is an idempotent in . The monoid underlying the algebra is again equal to with multiplication given by composition of relations (when a subgroup of is interpreted as a relation between and ). The algebras and are isomorphic via M?bius inversion in the poset . As an application we improve results by Bouc on the parametrization of simple modules of and also of simple biset functors, by using results of Linckelmann and Stolorz on the parametrization of simple modules of finite category algebras. Finally, in the case where is a cyclic group of order , we give an explicit isomorphism between and a direct product of matrix rings over group algebras of the automorphism groups of cyclic groups of order , where divides?.