The algebra of essential relations on a finite set

Serge Bouc (1) , Jacques Thévenaz (2)

1 LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée - UMR CNRS 7352
2 EPFL - Ecole Polytechnique Fédérale de Lausanne

Let X be a finite set and let k be a commutative ring. We consider the k-algebra of the monoid of all relations on X, modulo the ideal generated by the relations factorizing through a set of cardinality strictly smaller than Card(X), called inessential relations. This quotient is called the essential algebra associated to X. We then define a suitable nilpotent ideal of the essential algebra and describe completely the structure of the corresponding quotient, a product of matrix algebras over suitable group algebras. In particular, we obtain a description of the Jacobson radical and of all the simple modules for the essential algebra.

Domaines: Théorie des représentations [math.RT]; Anneaux et algèbres [math.RA]; Théorie des groupes [math.GR]

Serge Bouc, Jacques Thévenaz. The algebra of essential relations on a finite set. 2013. ⟨hal-00877555⟩