The $A$-fibered Burnside ring as $A$-fibered biset functor in characteristic zero

TL;DR

This paper studies the algebraic structure of the $A$-fibered Burnside ring functor over a field of characteristic zero, extending previous results to a broader class of groups and functors.

Contribution

It establishes foundational properties of the $A$-fibered Burnside ring functor as an $A$-fibered biset functor, including subfunctor lattice and composition factors, generalizing prior work.

Findings

Determined the lattice of subfunctors of $B_{ extbf{K}}^A$

Identified the composition factors of $B_{ extbf{K}}^A$

Extended results to broader classes of groups and functors

Abstract

Let $A$ be an abelian group such that $\mathrm{Hom}(G,A)$ is finite for all finite groups $G$, 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 foundational properties of the $A$-fibered Burnside ring functor $B_{\mathbb{K}}^A$ as an $A$-fibered biset functor over $\mathbb{K}$. This includes the determination of the lattice of subfunctors of $B_{\mathbb{K}}^A$ and the determination of the composition factors of $B_{\mathbb{K}}^A$. The results of the paper extend results of Co\c{s}kun and Y\i lmaz 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 over fields of characteristic zero.