Monomial $G$-posets and their Lefschetz invariants

Abstract

Let $G$ be a finite group, and $C$ be an abelian group. We introduce the notions of $C$-monomial $G$-sets and $C$-monomial $G$-posets, and state some of their categorical properties. This gives in particular a new description of the $C$-monomial Burnside ring $B_C(G)$. We also introduce Lefschetz invariants of $C$-monomial $G$-posets, which are elements of $B_C(G)$. These invariants allow for a definition of a generalized tensor induction multiplicative map $\\mathcal{T}_{U,\\lambda}: B_C(G)\\to B_C(H)$ associated to any $C$-monomial $(G,H)$-biset $(U,\\lambda)$, which in turn gives a group homomorphism $B_C(G)^\\times\\to B_C(H)^\\times$ between the unit groups of $C$-monomial Burnside rings.