Simple functors (Q1880138)

In order to give a uniform description for some operations (as the induction, the restriction, the inflation, the deflation) using the formalism for bi-sets, it is natural to consider a commutative ring \(k\), a class \(\mathcal{C}\) of finite groups and categories which have as class of objects the class \(\mathcal{C}\) and as homomorphisms some linear combinations with coefficients in \(k\). Then the category of \(k\)-linear functors from such a category into the category of \(k\)-modules is an abelian category and it is useful to establish connections between groups in \(\mathcal{C}\) and special objects (as simple, injective, projective objects) in this category. In the present paper \(k\) is a field of characteristic \(p\) and \(\mathcal{C}_{k}\) is the category which has as objects the finite \(p\)-groups and as homomorphisms the \(k\)-linear combinations of bi-sets. The author considers specific subfunctors of the Burnside functor which have a unique simple quotient and give some estimations for the \(k\)-dimension of the evaluation of these simple functors (proposition 3 and proposition 4). In the abelian case these estimations become equalities (corollary 2).