# Universal skein theory for group actions

## TL;DR

This paper develops a universal skein theory for group-action planar algebras, linking tensor network diagrams and subfactor theory, and answers a longstanding question by Vaughan Jones.

## Contribution

It introduces a universal skein framework for group-action planar algebras, unifying tensor diagrams and subfactor theory, and resolves a question posed by Vaughan Jones.

## Key findings

- Established a universal skein theory for group-action planar algebras.
- Connected group-action models with subfactor planar algebras.
- Provided a positive answer to Vaughan Jones's question from the 1990s.

## Abstract

Given a group action on a finite set, we define the group-action model which consists of tensor network diagrams which are invariant under the group symmetry. In particular, group-action models can be realized as the even part of group-subgroup subfactor planar algebras. Moreover, all group-subgroup subfactor planar algebras arise in this way from transitive actions. In this paper, we provide a universal skein theory for those planar algebras. With the help of this skein theory, we give a positive answer to a question asked by Vaughan Jones in the late nineties.