The monoidal structure of the category of partial representations of finite groups

TL;DR

This paper explores the monoidal structure of the category of partial representations of finite groups, revealing new descriptions of simple objects, tensor products, and subgroup relationships within this categorical framework.

Contribution

It introduces novel descriptions of simple objects and tensor products in the category of partial representations, and establishes embedding theorems relating subgroup and group partial representation categories.

Findings

The category of partial representations forms a multifusion category.

Simple objects and tensor products are characterized through an alternative description.

Partial representation categories of subgroups embed into those of the entire group for finite abelian groups.

Abstract

In this work, we analyze the structure of the category of partial representations of a finite group $G$ as a multifusion category, providing an alternative way to describe simple objects and their tensor products. We describe the interconnection between the category of partial representations of a finite group and the category of global representations of its subgroups (the Christmas Tree's Theorem). Also, for a finite abelian group $G$, we prove that the category of partial representations of any of its subgroups can be embedded into the category of partial representations of $G$ (the Matryoshka's Theorem).