Group partition categories

TL;DR

This paper introduces the group partition category associated with a group G, providing explicit bases, presentations, and an embedding into the group Heisenberg category, linking it to wreath product categories.

Contribution

It constructs the group partition category, establishes its embedding into the group Heisenberg category, and identifies its Karoubi envelope with a wreath product interpolating category.

Findings

Explicit bases for morphism spaces

Efficient presentation via generators and relations

Equivalence of the Karoubi envelope to a wreath product interpolating category

Abstract

To every group $G$ we associate a linear monoidal category $\mathcal{P}\mathit{ar}(G)$ that we call a group partition category. We give explicit bases for the morphism spaces and also an efficient presentation of the category in terms of generators and relations. We then define an embedding of $\mathcal{P}\mathit{ar}(G)$ into the group Heisenberg category associated to $G$. This embedding intertwines the natural actions of both categories on modules for wreath products of $G$. Finally, we prove that the additive Karoubi envelope of $\mathcal{P}\mathit{ar}(G)$ is equivalent to a wreath product interpolating category introduced by Knop, thereby giving a simple concrete description of that category.