A generating problem for subfactors

TL;DR

This paper proves that a family of group-subgroup subfactors associated with Kneser graphs can be generated by 2-boxes, answering a question of Vaughan Jones and linking to quantum permutation groups and strongly regular graphs.

Contribution

It establishes that these subfactors are generated by 2-boxes, providing a minimal generating set and connecting subfactor theory with quantum symmetries and graph theory.

Findings

Subfactors associated with Kneser graphs are generated by 2-boxes.

The result answers Vaughan Jones's question on minimal generators.

Identifies an infinite family of strongly regular graphs with no quantum symmetry.

Abstract

Bisch and Jones proposed the classification of planar algebras by simple generators and relations. In this paper, we study the generating problem for a family of group-subgroup subfactors associated with the Kneser graphs, namely, to determine the generators with minimal size. In particular, we prove that this family of subfactors are generated by $2$-boxes and this provides an affirmative answer to a question of Vaughan Jones. This generator problem is also related to the theory of quantum permutation groups, and the main theorem also provides an infinite family of strongly regular graphs with no quantum symmetry.