Fusion categories via string diagrams

TL;DR

This paper employs string diagram calculus to provide graphical proofs of fundamental properties of fusion categories, including duality, positivity, and rigidity, and introduces a new pairing convention for graphical computations.

Contribution

It introduces the pairing convention as a new graphical framework and characterizes pivotal structures via explicit algebraic equations, advancing the graphical understanding of fusion categories.

Findings

Graphical proofs of duality and rigidity in fusion categories

Introduction of the pairing convention for string diagrams

Explicit algebraic characterization of pivotal structures

Abstract

We use the string diagram calculus to give graphical proofs of the basic results of Etingof, Nikshych and Ostrik on fusion categories. These results include: the quadruple dual is canonically isomorphic to the identity, positivity of the paired dimensions, and Ocneanu rigidity. We introduce the pairing convention as a convenient graphical framework for working with fusion categories. We use this framework to express the pivotal operators as a product of the apex associator monodromy and the pivotal indicators. We also characterize pivotal structures as solutions of an explicit set of algebraic equations over the complex numbers.