The extension algebra of some cohomological Mackey functors (Q2517215)

Given a field \(k\) of positive characteristic \(p\), the authors investigate cohomological Mackey functors for finite groups over \(k\). Their main result is a presentation by generators and relations of the graded algebra of self-extensions \(\mathcal E=\text{Ext}_{\text M^c_k(G)}(S_1^G,S_1^G)\) of a particular simple functor \(S_1^G\) of the Mackey algebra M\({}^c_k(G)\) of a given finite group \(G\) (of order divisible by \(p\)), when \(p\) is odd and \(G\) elementary abelian. Their proof mainly builds on previous results of the first author who answered the question for \(p=2\) [\textit{S. Bouc}, Adv. Math. 221, No. 3, 983--1045 (2009; Zbl 1194.20050)], and it also relies on a certain inflation functor \(\sigma_{G/N}^G:\text M^c_k(G/N)\to\text M^c_k(G)\) which is exact for any finite group \(G\) and normal subgroup \(N\). Furthermore, the authors prove a conjecture about the Poincaré series of \(\mathcal E\), which was proved by the first author in the case \(p=3\).