Multiple Burnside rings and Brauer induction formulae

Abstract

Let G be a finite group, and suppose that G is an operator group of a finite group A . Define S ( G , A ) = { ( H , σ ) | H ∈ S ( G ) and σ ∈ Z 1 ( H , A ) } , where S ( G ) is the set of subgroups of G and Z 1 ( H , A ) is the set of crossed homomorphisms from H to A . We view G as an operator group of the opposite group A ○ of A , and make S ( G , A ) into a left A ○ ⋊ G -set. The ring Ω ( G , A ) is defined to be a commutative ring consisting of all formal Z -linear combinations of A ○ ⋊ G -orbits in S ( G , A ) . Idempotent formulae for Q ⊗ Z Ω ( G , A ) not only imply a generalization of Dress' induction theorem but bring, in the case where Z 1 ( G , A ) is the set of linear C -characters of G , Boltje's explicit formula for Brauer's induction theorem and its hyperelementary version.