On fibred biset functors with fibres of order prime and four

TL;DR

This paper presents a counterexample to a conjecture on simple modules over Green biset functors and classifies simple modules for the monomial Burnside ring over prime order groups.

Contribution

It provides a counterexample to a previous conjecture and classifies simple modules for specific monomial Burnside rings.

Findings

Counterexample to the conjecture for cyclic group of order four

Classification of simple modules over prime order groups

Validation of the parametrization in certain cases

Abstract

This note has two purposes: First, to present a counterexample to a conjecture parametrizing the simple modules over Green biset functors, appearing in an author's previous article. This parametrization fails for the monomial Burnside ring over a cyclic group of order four. Second, to classify the simple modules for the monomial Burnside ring over a group of prime order, for which the above-mentioned parametrization holds.