The global representation fibered ring

TL;DR

This paper introduces the global representation fibered ring of a finite group, analyzing its algebraic structure, homomorphisms, spectrum, and primitive idempotents to deepen understanding of fibered biset functors.

Contribution

It defines and studies the global representation fibered ring, combining fibered Burnside and character rings, and computes its algebraic properties and structure.

Findings

Computed all ring homomorphisms to complex numbers

Determined the spectrum and connected components

Identified primitive idempotents over  and conductors

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.