Diagrammatic Categories which arise from Representation Graphs
Ryan Reynolds
TL;DR
This work develops a general framework to produce diagrammatic categories from representation graphs $R(V,G)$ (the McKay quivers) of a group $G$, yielding a diagrammatic category $\mathbf{Dgrams}_{R(V,G)}$ whose quotient can be equivalent to the full monoidal subcategory of $G$-mod generated by tensor products of irreducible modules. By introducing the categories $G$-mod$_{irr}$ and $\mathbf{Dgrams}_{R(V,G)}$ and defining the functors $\mathcal{H}_{R(V,G)}$ and its quotient $\overline{\mathcal{H}}$, the paper proves fullness, and under explicit hypotheses, faithfulness and categorical equivalence, thereby providing diagrammatic presentations of irreducible-module subcategories. It highlights concrete instances, including the McKay correspondence with affine Dynkin diagrams, the Temperley–Lieb paradigm for $SU(2)$, and extensions to fusion categories such as Ver$_p$ and Fibonacci categories, illustrating the method's breadth. The results offer a versatile bridge between combinatorial graph data and monoidal/diagrammatic presentations, with potential impact on categorical representation theory and decategorification, and they outline avenues for generalization to broader graph types and fusion frameworks.
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.