Permutation, stabilization and decomposition

TL;DR

This paper explores the stable permutation category of finite groups, analyzing its structure and decomposition, particularly focusing on cyclic and generalized quaternion groups based on tt-geometry insights.

Contribution

It provides a proper definition of the stable permutation category and proves its decomposition exclusively over cyclic and generalized quaternion groups.

Findings

The stable permutation category decomposes over cyclic groups.

It also decomposes over generalized quaternion groups.

No other group types exhibit this decomposition.

Abstract

Informed by our understanding of the tt-geometry of permutation modules, we investigate the proper definition of the `stable permutation category' of a finite group. Then we prove that this category decomposes over cyclic and generalized quaternion groups and only in those cases.