Appears in collection : Algebraic and finite groups, geometry and representations. 50 years of Chevalley seminar / Groupes algébriques, groupes finis, géométries, représentations : 50 ans de séminaire Chevalley.

In this joint work with Jacques Thévenaz, we develop the representation theory of finite sets and correspondences : let kC the category of finite sets, in which morphisms are k-linear combinations of correspondences (where k is a given commutative ring), and let Fk be the category of correspondence functors (over k), i.e. the category of k-linear functors from kC to k-modules. This category Fk is an abelian k-linear category. In such a framework, it is of crucial importance to describe the algebra of essential endomorphisms of a given object. This is what we achieved in a previous work on the algebra of essential relations on a finite set, describing in particular its simple modules. This description leads to a parametrization of the simple functors on kC by triples (E;R;V) consisting of a finite set E, a partial order relation R on E, and a simple k-linear representation V of the automorphism group of (E;R).