The slice Burnside ring and the section Burnside ring of a finite group

TL;DR

This paper introduces the slice and section Burnside rings for finite groups, extending classical results and establishing their structure as Green biset functors, with implications for their unit groups.

Contribution

It defines two new Burnside rings based on morphisms and Galois morphisms of G-sets, extending known properties of the classical Burnside ring.

Findings

Extended ghost map and prime spectrum descriptions to new rings

Established Green biset functor structures for the rings

Discussed the functorial properties of their unit groups

Abstract

This paper introduces two new Burnside rings for a finite group $G$, called the slice Burnside ring and the section Burnside ring. They are built as Grothendieck rings of the category of morphisms of $G$-sets, and of Galois morphisms of $G$-sets, respectively. The well known results on the usual Burnside ring, concerning ghost maps, primitive idempotents, and description of the prime spectrum, are extended to these rings. It is also shown that these two rings have a natural structure of Green biset functor. The functorial structure of unit groups of these rings is also discussed.