Type $G_2$ Quantum Subgroups from Graph Planar Algebra Embeddings

TL;DR

This paper introduces a graphical method to identify quantum subgroups of type G2 by embedding their planar algebras into graph planar algebras, enabling computational discovery of algebraic relations.

Contribution

It presents a novel graphical embedding technique for quantum subgroups of type G2 using graph planar algebras, facilitating relation discovery.

Findings

Explicit graphical presentations for G2 quantum subgroups

New relations uncovered through computational methods

Method applicable to other tensor categories

Abstract

We give graphical presentations for the two quantum subgroups of type $G_2$. To do this we use a method of extending a tensor category by embedding the planar algebra of a $\otimes$-generating object into the graph planar algebra of this object's fundamental graph. This allows the use of computational methods to uncover relations we would have little hope of arriving at otherwise.