The A-fibered Burnside Ring as A-Fibered Biset Functor in Characteristic Zero

Let A be an abelian group such that torn(A) is finite for every n ≥ 1 and let ${\mathbb{K}}$ be a field of characteristic zero containing roots of unity of all orders equal to finite element orders in A. In this paper we prove fundamental properties of the A-fibered Burnside ring functor $B_{\mathbb{K}}^{A}$ as an A-fibered biset functor over K. This includes a description of the composition factors of $B^{A}_{\mathbb{K}}$ and the lattice of subfunctors of $B_{\mathbb{K}}^{A}$ in terms of what we call BA-pairs and a poset structure on their isomorphism classes. Unfortunately, we are not able to classify BA-pairs. The results of the paper extend results of Coşkun and Yılmaz for the A-fibered Burnside ring functor restricted to p-groups and results of Bouc in the case that A is trivial, i.e., the case of the Burnside ring functor as a biset functor over fields of characteristic zero. In the latter case, BA-pairs become Bouc’s B-groups which are also not known in general.