Fibred permutation sets and the idempotents and units of monomial Burnside rings
Let \(G\) be a finite group and \(A\) an abelian group. An \(A\)-fibred \(G\)-set is an \(A\)-free \(A\times G\)-set with only finitely many orbits. The category of \(A\)-fibred \(G\)-sets is with products and coproducts, and the corresponding Grothendieck ring \(B(A,G)\) is called the monomial Burnside ring for \(G\) with fibre group \(A\). The author mainly considers the case when the fibre group is a supercyclic group \(C\), that is, isomorphic to a subgroup of \({\mathbb Q}/{\mathbb Z}\). An important part of the paper is devoted to the study of the idempotents of the monomial Burnside algebra \(K\otimes_{\mathbb Z}B(C,G)\), where \(K\) is a field of characteristic zero, by employing a monomial version of the Möbius inversion. The main result of the paper is concerned with the group of units \(B(C,G)^*\) of the monomial Burnside ring. The author constructs a tenduction map, and proves that the functor \(B(C,-)^*\) defined on the category of inclusions of finite groups, and equipped with the tenduction, restriction and conjugation maps, is a Mackey functor over \(\mathbb Z\).
scientific article; zbMATH DE number 2119113
scientific article