Around evaluations of biset functors

Abstract

The evaluation functor carries important information about the category of biset functors into the category of modules over the double Burnside algebra. Our purpose here, is to study double Burnside algebras via evaluations of biset functors. In order to avoid the difficult problem of vanishing of simple functors, we look at finite groups for which there is no non-trivial vanishing and we call them non-vanishing groups. This family contains all the abelian groups, but also infinitely many others. We show that for a non-vanishing group, there is an equivalence between the category of modules over the double Burnside algebra and a specific category of biset functors. Then, we deduce results about the highest-weight structure, and the self-injective property of the double Burnside algebra. We also revisit Barker's Theorem on the semi-simplicity of the category of biset functors.

Semisimplicity of some deformations of the subgroup category and the biset category

“…Surprisingly, for arbitrary K, K, ℓ, it is not hard to classify the simple Λ-modules. That is in contrast to the situation for KB K , where a suitable closure condition has to be imposed on K to avoid the "vanishing problem" discussed in Rognerud [Rog19]. In Section 3, we shall show that, for Λ, the "vanishing problem" itself vanishes.…”

“…Several examples in Bouc-Stancu-Thévenaz [BST13, Section 13] show that, for biset functors, there are no direct analogues of the latest lemma or the two theorems below in this section. See also the discussion of the "vanishing problem" in Rognerud [Rog19]. Proof.…”