The primitive idempotents of the \(p\)-permutation ring (Q975095)

Let \(G\) be a finite group, \(k\) a field of characteristic \(p>0\), and \(K\) a field of characteristic \(0\). The fields \(k\) and \(K\) are assumed to be `large enough' in a precise sense. Denote by \(pp_k(G)\) the Grothendieck ring (w.r.t. direct sum decompositions and tensor products over \(k\)) of \(p\)-permutation \(kG\)-modules. In this paper, the authors obtain explicit formulae for the primitive idempotents of \(K\otimes_{\mathbb Z}pp_k(G)\), the motivation coming from applications to Cartan matrices of Mackey algebras. A key ingredient is a natural ring homomorphism from the Burnside ring \(B(G)\) to \(pp_k(G)\), along with the classical formulae for the primitive idempotents of \(K\otimes_{\mathbb Z}B(G)\).