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$p$-Bifree biset functors

Olcay Coşkun, Deniz Yılmaz

TL;DR

This work defines and analyzes $p$-bifree biset functors, a category sitting between classical biset functors and diagonal $p$-permutation functors by restricting to bisets with $p'$-stabilizers on both sides. It proves a complete classification of simple $p$-bifree biset functors: $S^{ riangle}_{G,V}$ for finite $G$ and simple $K ext{Out}(G)$-modules $V$, and computes composition factors for key functors, including the Burnside ring, Brauer character ring, and ordinary character ring, with explicit decompositions in several cases (e.g., $S_{C_p imes C_p,inom{ ext{C}}{ ext{C}}}$ and $S_{C_q imes C_q,inom{ ext{C}}{ ext{C}}}$ for $q eq p$). The paper develops the structure theory of $p$-bifree functors via the essential algebras $ ext{E}^{ riangle}(G)\cong ext{Out}(G)$, subfunctor lattices, and restriction kernels, and links these to classical representation-theoretic objects through precise short exact sequences and semisimplicity results. Overall, it constructs a bridge linking traditional biset functor theory with diagonal $p$-permutation phenomena and provides detailed, computable descriptions of the simple constituents across several fundamental functors. This framework clarifies how modular and ordinary representation theories integrate within the $p$-bifree context and yields concrete decompositions of well-known functors into elementary $S^{ riangle}_{G,V}$ components.

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 $q\neq p$.

$p$-Bifree biset functors

TL;DR

This work defines and analyzes -bifree biset functors, a category sitting between classical biset functors and diagonal -permutation functors by restricting to bisets with -stabilizers on both sides. It proves a complete classification of simple -bifree biset functors: for finite and simple -modules , and computes composition factors for key functors, including the Burnside ring, Brauer character ring, and ordinary character ring, with explicit decompositions in several cases (e.g., and for ). The paper develops the structure theory of -bifree functors via the essential algebras , subfunctor lattices, and restriction kernels, and links these to classical representation-theoretic objects through precise short exact sequences and semisimplicity results. Overall, it constructs a bridge linking traditional biset functor theory with diagonal -permutation phenomena and provides detailed, computable descriptions of the simple constituents across several fundamental functors. This framework clarifies how modular and ordinary representation theories integrate within the -bifree context and yields concrete decompositions of well-known functors into elementary components.

Abstract

We introduce and study the category of -bifree biset functors for a fixed prime , defined via bisets whose left and right stabilizers are -groups. This category naturally lies between the classical biset functors and the diagonal -permutation functors, serving as a bridge between them. Every biset functor and every diagonal -permutation functor restricts to a -bifree biset functor. We classify the simple -bifree biset functors over a field of characteristic zero, showing that they are parametrized by pairs , where is a finite group and is a simple -module. As key examples, we compute the composition factors of several representation-theoretic functors in the -bifree setting, including the Burnside ring functor, the -bifree Burnside functor, the Brauer character ring functor, and the ordinary character ring functor. We further investigate classical simple biset functors, and for a prime .

Paper Structure

This paper contains 10 sections, 30 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Category of $p$-free bisets
  3. $p$-bifree biset functors
  4. Burnside ring as a $p$-bifree biset functors
  5. The functor $\mathbb{K}B^{\Delta}$
  6. The simple functor $S_{1,\mathbb{K}}^\Delta$
  7. The functor of Brauer characters
  8. The functor of complex character ring
  9. Further examples
  10. Appendix

Key Result

Lemma 4.1

Let $F$ be a $p$-bifree biset subfunctor of $RB$. Then for any finite group $G$, the $R$-module $F(G)$ is an ideal of $RB(G)$.

Theorems & Definitions (43)

  • Definition 2.1
  • Definition 3.1
  • Remark 3.2
  • Definition 3.3
  • Lemma 4.1
  • Proposition 4.2
  • Definition 4.3
  • Remark 4.4
  • Theorem 4.5
  • Theorem 4.6
  • ...and 33 more