Diagrammatic Categories which arise from Directed Graphs

Abstract

There is a categorical equivalence between the Temperley--Lieb category $TL(2)$ and the full subcategory of $SU(2)$-\textbf{mod} with objects given by $V^{\otimes k}$ where $V$ is the tautological $SU(2)$-module and $k$ is a non-negative integer. The first results in this dissertation develop new diagrammatic categories which are shown to be equivalent to similarly defined full subcategories of $G$-\textbf{mod} for certain finite subgroups $G$ of $SU(2)$. The diagrams which generate the Temperley--Lieb category are shown to be linear combinations of the generating diagrams for these newly defined diagrammatic categories. 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.