On the direct product of partial Burnside rings

TL;DR

This paper investigates the structure of the direct product of partial Burnside rings, revealing how the unit group of such rings for Coxeter groups decomposes into products related to irreducible components.

Contribution

It provides a detailed structural description of the direct product of partial Burnside rings, especially for Coxeter groups, and establishes an isomorphism of unit groups.

Findings

Unit group of partial Burnside ring for Coxeter groups decomposes into products.

Structural description of direct product of partial Burnside rings.

Isomorphism between unit groups of reducible and irreducible Coxeter groups.

Abstract

In this paper, we describe the structure of the direct product of partial Burnside rings of relative to the collection of a finite group. In particular, we show that the unit group of the partial Burnside ring relative to the set of all parabolic subgroups of a reducible finite Coxeter group is isomorphic to the direct product of unit groups of the partial Burnside ring of irreducible Coxeter group.