Rational $p$-biset functors

Abstract

In this paper, I give several characterizations of {\em rational biset functors over $p$-groups}, which are independent of the knowledge of genetic bases for $p$-groups. I also introduce a construction of new biset functors from known ones, which is similar to the Yoneda construction for representable functors, and to the Dress construction for Mackey functors, and I show that this construction preserves the class of rational $p$-biset functors.\par This leads to a characterization of rational $p$-biset functors as additive functors from a specific quotient category of the biset category to abelian groups. Finally, I give a description of the largest rational quotient of the Burnside $p$-biset functor : when $p$ is odd, this is simply the functor $R_\Q$ of rational representations, but when $p=2$, it is a non split extension of $R_\Q$ by a specific uniserial functor, which happens to be closely related to the functor of units of the Burnside ring.