A ghost algebra of the double Burnside algebra in characteristic zero

Abstract

For a finite group $G$, we introduce a multiplication on the \QQ-vector space with basis \scrS_{G\times G}, the set of subgroups of $G\times G$. The resulting \QQ-algebra \Atilde 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 \Atilde is a ring homomorphism. Our approach interprets \QQ 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 \scrS_{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 \Atilde are isomorphic via M\"obius inversion in the poset \scrS_{G\times G}. As an application we improve results by Bouc on the parametrization of simple modules of \QQ 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 \QQ 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 $n$.Comment: 41 pages. Changed title from "Ghost algebras of double Burnside algebras via Schur functors" and other minor changes. Final versio