On fibered Burnside rings, fiber change maps and cyclic fiber groups

TL;DR

This paper explores fibered Burnside rings, introducing fiber change maps and analyzing their properties, with a focus on conductors for cyclic fiber groups and their applications in representation theory.

Contribution

It introduces fiber change maps between fibered Burnside rings and studies their functoriality, providing new results on conductors for cyclic fiber groups.

Findings

Fiber change maps are functorial and natural with respect to biset operations.

Conductors for cyclic fiber groups are advanced and fully determined in key cases.

The work covers a wide range of examples illustrating the theory.

Abstract

Fibered Burnside rings appear as Grothendieck rings of fibered permutation representations of a finite group, generalizing Burnside rings and monomial representation rings. Their species, primitive idempotents and their conductors are of particular interest in representation theory as they encode information related to the structure of the group. In this note, we introduce fiber change maps between fibered Burnside rings, and we present results on their functoriality and naturality with respect to biset operations. We present some advances on the conductors for cyclic fiber groups, and fully determine them in particular cases, covering a wide range of interesting examples.