Ghost Algebras of Double Burnside Algebras via Schur Functors

Abstract

For 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 $n$.