$p$-Biset Functor of Monomial Burnside Rings

TL;DR

This paper analyzes the structure of the monomial Burnside biset functor over characteristic zero fields, explicitly describing restriction kernels, composition factors, and their minimal groups, providing rare complete classifications.

Contribution

It offers a detailed classification of composition factors of the monomial Burnside biset functor, including explicit descriptions and minimal groups, a rare comprehensive example.

Findings

Explicit description of restriction kernels at each finite p-group G.

Complete list of composition factors of the functor.

Identification of simple modules as evaluations of composition factors.

Abstract

We investigate the structure of the monomial Burnside biset functor over a field of characteristic zero, with particular focus on its restriction kernels. For each finite \( p \)-group \( G \), we give an explicit description of the restriction kernel at \( G \), and determine the complete list of composition factors of the functor. We prove that these composition factors have minimal groups \( H \) isomorphic either to a cyclic \( p \)-group or to a direct product of such a group with a cyclic group of order \( p \). Furthermore, we identify the simple \( \mathbb{C}[\Aut(H)] \)-modules that appear as evaluations of these composition factors at their minimal groups. Explicit classifications of composition factors for biset functors are rare, and our results provide one of the few complete examples of such classifications.