Extension groups between simple Mackey functors
A Mackey functor for a finite group \(G\) over an algebraically closed field \(k\) of characteristic \(p>0\) can be viewed as a module over the Mackey algebra \(\mu_k(G)\). For every subgroup \(H\) of \(G\) and every simple \(kN_G(H)/H\)-module \(V\), there is a simple Mackey funktor \(S_{H,V}\) for \(G\) over \(k\). In the paper under review, the author investigates the groups \(\text{Ext}^1_{\mu_k(G)}(S_{Q,W}, S_{H,V})\), for subgroups \(Q,H\) of \(G\), a simple \(kN_G(H)/H\)-module \(V\) and a simple \(kN_G(Q)/Q\)-module \(W\). One of her main results is concerned with the case where \(Q\) and \(H\) are distinct normal subgroups of \(G\). Another deals with the case where \(Q\) and \(H\) are non-conjugate, and \(V,W\) are both trivial. A third result handles the case where \(G\) has a normal Sylow \(p\)-subgroup and \(Q,H\) are non-conjugate. Finally, the author relates \(\text{Ext}^1_{\mu_k(G)}(S_{H,V}, S_{H,W})\) and \(\text{Ext}^1_{N_G(H)/H}(V,W)\) when either \(V,W\) are both trivial, or \(G\) has a normal Sylow \(p\)-subgroup.
scientific article; zbMATH DE number 5545851
scientific article