# Monomial $G$-posets and their Lefschetz invariants

## TL;DR

This paper introduces $C$-monomial $G$-sets and posets, describes their categorical properties, and develops Lefschetz invariants that lead to new homomorphisms between unit groups of $C$-monomial Burnside rings.

## Contribution

It provides a new categorical framework for $C$-monomial $G$-sets and posets, and constructs Lefschetz invariants that induce group homomorphisms between Burnside ring units.

## Key findings

- Defined $C$-monomial $G$-sets and posets with categorical properties
- Established a new description of the $C$-monomial Burnside ring $B_C(G)$
- Constructed Lefschetz invariants leading to homomorphisms between unit groups

## 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.