$p$-Bifree biset functors

TL;DR

This paper introduces the category of p-bifree biset functors, classifies their simple objects over characteristic zero fields, and computes composition factors of key representation-theoretic functors within this framework.

Contribution

It defines p-bifree biset functors, classifies their simple modules, and analyzes their relation to classical biset and p-permutation functors, providing new insights into their structure.

Findings

Classified simple p-bifree biset functors over characteristic zero fields.

Computed composition factors of key representation-theoretic functors in the p-bifree setting.

Analyzed classical simple biset functors for specific groups.

Abstract

We introduce and study the category of $p$-bifree biset functors for a fixed prime $p$, defined via bisets whose left and right stabilizers are $p'$-groups. This category naturally lies between the classical biset functors and the diagonal $p$-permutation functors, serving as a bridge between them. Every biset functor and every diagonal $p$-permutation functor restricts to a $p$-bifree biset functor. We classify the simple $p$-bifree biset functors over a field $K$ of characteristic zero, showing that they are parametrized by pairs $(G,V)$, where $G$ is a finite group and $V$ is a simple $K\mathrm{Out}(G)$-module. As key examples, we compute the composition factors of several representation-theoretic functors in the $p$-bifree setting, including the Burnside ring functor, the $p$-bifree Burnside functor, the Brauer character ring functor, and the ordinary character ring functor. We further investigate classical simple biset functors, $S_{C_p \times C_p, \mathbb{C}}$ and $S_{C_q \times C_q, \mathbb{C}}$ for a prime $q\neq p$.