# Ore's theorem on subfactor planar algebras

## TL;DR

This paper generalizes Ore's theorem from finite groups to subfactor planar algebras, establishing a link between combinatorics and representation theory through the structure of biprojection lattices.

## Contribution

It proves that distributive biprojection lattices in irreducible subfactor planar algebras admit minimal 2-box projections generating the identity, extending Ore's theorem to a new mathematical context.

## Key findings

- Distributive biprojection lattices imply the existence of minimal 2-box projections.
- Establishes a connection between subfactor theory and combinatorial group theory.
- Generalizes a classical group-theoretic theorem to the setting of operator algebras.

## Abstract

This article proves that an irreducible subfactor planar algebra with a distributive biprojection lattice admits a minimal 2-box projection generating the identity biprojection. It is a generalization (conjectured in 2013) of a theorem of Oystein Ore on distributive intervals of finite groups (1938), and a corollary of a natural subfactor extension of a conjecture of Kenneth S. Brown in algebraic combinatorics (2000). We deduce a link between combinatorics and representations in finite group theory.