On the occasion of his retirement, we celebrate the work of Serge Bouc
Double Burnside algebras and quasi-hereditary algebras
A quasi-hereditary algebra is a pair consisting of an algebra and a partial order on the isomorphism classes of its simple modules satisfying conditions coming from the representation theory of semisimple complex Lie algebras. By a famous result of Peter Webb, we know that the category of biset functors over a field of characteristic zero has a similar behavior and we will see how this can be used to study the quasi-hereditary property of the double Burnside algebra. We should remark that the definition of quasi-hereditary algebra involves the choice of a partial order, hence an algebra is in general not `canonically' quasi-hereditary. We will see in a simple example that in general there is a large number of such partial orders.