The global representation fibered ring
J. Miguel Calderón, Alberto G. Raggi-Cárdenas
TL;DR
This work constructs the global $A$-fibered representation ring Д^A(G) from $A$-fibered Burnside data and $X/A$-graded $\mathbb{C}G$-modules, establishing a concrete basis, a Mackey-type product, and a finiteness criterion. It proves that all ring homomorphisms to $\mathbb{C}$ (the marks) are given by a complete, invertible table of species $S_{H,\varphi,b}$, and it provides an explicit decomposition into primitive idempotents over $\mathbb{Q}[\omega]$ via Möbius inversion, along with conductor calculations. The prime spectrum of Д^A(G) is described as a finite union of connected components indexed by perfect subgroups and conjugacy classes, and the idempotent structure ties into solvability questions, with a Feit–Thompson-type conduit. Finally, Д^A is shown to be a fibered-biset functor with explicit action under standard operations, enabling representation-theoretic applications and illustrating deep links between algebraic structure and group solvability.
Abstract
In this paper, we combine the concepts of the fibered Burnside ring and the character ring, viewing them as fibered biset functors, into what we call the global representation fibered ring of a finite group. We compute all ring homomorphisms from this ring to the complex numbers, determine its spectrum and its connected components, and identify the primitive idempotents of this ring tensor with $\mathbb{Q}$ and its conductors.