$p$-Biset Functor of Monomial Burnside Rings

TL;DR

The paper analyzes the monomial Burnside biset functor over a field of characteristic zero, focusing on the kernel $\mathbb S_p$ of the monomial linearization map and its composition factors. Using BCK, it reduces the task to determining restriction kernels $\mathcal{K}\mathbb S_p(G)$ for finite $p$-groups and then identifies the corresponding $\mathbb{C}[\operatorname{Out}(G)]$-modules that occur. The authors provide explicit, complete classifications of composition factors for several families of $p$-groups: cyclic $p$-groups, $C_p \times C_p$, and mixed abelian cases $C_{p^m} \times C_p$ (including the even-$p$ and odd-$p$ settings). They also describe the $\mathbb{C}[\operatorname{Aut}(G)]$-structure of these kernels, compute indecomposable summands via Mackey and Clifford theories, and present concrete examples for small primes. Overall, the work delivers one of the few comprehensive classifications of composition factors for a biset functor and lays groundwork for a full indecomposable decomposition in future work.

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.