Diagrammatic Categories which arise from Representation Graphs

TL;DR

This paper constructs a diagrammatic category from a group's representation graph and proves an equivalence with a subcategory of group modules, providing a new categorical perspective on group representations.

Contribution

It introduces a general construction of diagrammatic categories from representation graphs and establishes an equivalence with certain subcategories of group modules.

Findings

Construction of $ extbf{Dgrams}_{R(V,G)}$ from representation graphs.

Proof of categorical equivalence under explicit criteria.

Provides a new diagrammatic framework for understanding group representations.

Abstract

The main result of this paper utilizes the representation graph of a group $G$, $R(V,G)$, and gives a general construction of a diagrammatic category $\mathbf{Dgrams}_{R(V,G)}$. The proof of the main theorem shows that, given explicit criteria, there is an equivalence of categories between a quotient category of $\mathbf{Dgrams}_{R(V,G)}$ and a full subcategory of $G-\textbf{mod}$ with objects being the tensor products of finitely many irreducible $G$-modules.