Relative \(B\)-groups (Q2283344)

Let \(G\) be a finite group and \(\mathbb{F}B\) be the Burnside functor, where \(\mathbb{F}\) is a field of characteristic zero, this is a Green biset functor. Given a fixed finite group \(K\), the shifted biset functor \(B_{K}\) is given by \(B_{K}(G)=B(G\times K)\), this is again a biset functor and has a natural algebra structure from the one of \(B\), these give an algebra structure on \(\mathbb{F}B_{K}\) which is known to be semisimple with primitive idempotents denoted by \(e_{L}^{G\times L}\) where \(L\) runs on conjugacy classes of subgroups of \(G\times K\). The author fully describes the lattice of ideals of \(\mathbb{F}B_{K}\) and of its restriction to \(p\)-groups \(\mathbb{F}B_{K}^{(p)}\). The setup is as follows, let \(\mathbf{grp}_{\Downarrow K}\) be the category with objects pairs \((L,\varphi )\) where \(L\) is a finite group and \(\varphi :L\to K\) is a group homomorphism, morphisms are defined via commutative diagrams up to inner automorphisms of \(K\). An ideal in \(\mathbb{F}B_{K}\) is determined by certain idempotents, \(e_{L,\varphi}\), where \((L,\varphi) \) runs on certain isomorphisms classes, \(\mathcal{S}_{K}\). Moreover, a group over \(K\), \((L,\varphi)\), determines certain groups called \(B_{K}\)-groups that capture basic ideals of \(\mathbb{F}B_{K}\). Next, the author describes the idempotents that correspond to a \(B_{K}\)- group \((L,\varphi)\). The main result is a full description of the lattice of ideals of \(\mathbb{F}B_{K}\) in terms of the lattice of certain \(B_{K}\)- subgroups \((L,\varphi)\) of \(\mathbf{grp}_{\Downarrow K}\). The methods also give a similar description for the restriction of \(\mathbb{F}B_{K}\) to \(p\)-groups \(\mathbb{F}B_{K}^{(p)}\).