Around evaluations of biset functors

Abstract

For a non-vanishing group, we show that the evaluation functor induces an equivalence between the category of modules over the double Burnside algebra and a certain category of biset functors. Using this equivalence, we deduce that over a field of characteristic zero, the double Burnside algebra of a non-vanishing group is a quasi-hereditary algebra. We also show that the double Burnside algebra over a field is self-injective if and only if it is semisimple.

v2: there is now a part on the exact Borel subalgebras. I also gave a proof 'à la Bouc-Thévenaz'of Barker's Theorem on the semi-simplicity of the category of biset functors. v3: accepted for publication in Annales de l'institut Fourier