Green correspondence on centric Mackey functors over fusion systems (Q6066484)

Let \(G\) be a finite group, \(p\) a prime, \(\mathcal{R}\) a complete PID with residue field of characteristic \(p\). Given a \(p\)-subgroup \(V\) of \(G\), the Green correspondence states that there exists a bijection between the finitely generated indecomposable \(\mathcal{R}G\)-modules with vertex \(V\) and finitely generated \(\mathcal{R}N_G(V)\)-modules with vertex \(V\). A Mackey functor is an algebraic object for operations which behave similar to induction, restriction and conjugation are present and is defined over finite groups. The author provides a generalization of this concept by definining the notion of a centric Mackey functor over a fusion system (Definition \(2.29\)). The main result of the paper is that the Green correspondence holds for centric Mackey functors over fusion systems (Theorem \(4.37\)). The theorem establishes that there is a unique decomposition for certain centric Mackey functors, one of which is induced from an indecomposable one with vertex \(H\), into indecomposable ones. Moreover, it also states that in each of these decompositions, there exists a unique indecomposable summands with vertex \(H\).