Central idempotents of the bifree and left-free double Burnside ring
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.
scientific article; zbMATH DE number 6361331
scientific article