# Ore's theorem on cyclic subfactor planar algebras and beyond

## TL;DR

This paper generalizes Ore's theorem to cyclic subfactor planar algebras, showing they are singly generated and exploring related conjectures, dual theorems, and bounds in the context of subfactor theory and finite groups.

## Contribution

It extends Ore's theorem to subfactor planar algebras, proving that cyclic subfactors are singly generated and proposing conjectures for broader cases.

## Key findings

- Cyclic subfactor planar algebras are singly generated by a minimal 2-box projection.
- A dual version of Ore's theorem is established in the subfactor context.
- A non-trivial upper bound for the minimal number of irreducible components of a faithful complex representation is provided.

## Abstract

Ore proved that a finite group is cyclic if and only if its subgroup lattice is distributive. Now, since every subgroup of a cyclic group is normal, we call a subfactor planar algebra cyclic if all its biprojections are normal and form a distributive lattice. The main result generalizes one side of Ore's theorem and shows that a cyclic subfactor is singly generated in the sense that there is a minimal 2-box projection generating the identity biprojection. We conjecture that this result holds without assuming the biprojections to be normal, and we show that it is true for small lattices. We finally exhibit a dual version of another theorem of Ore and a non-trivial upper bound for the minimal number of irreducible components for a faithful complex representation of a finite group.