# Euler totient of subfactor planar algebras

## TL;DR

This paper generalizes Euler's totient function to subfactor planar algebras using the biprojection lattice's Mobius function, revealing structural properties and applications to finite groups and their representations.

## Contribution

It introduces a new Euler totient concept for subfactor planar algebras and explores its implications for group representations and biprojection structures.

## Key findings

- Nonzero Euler totient implies existence of a minimal 2-box projection
- Defines dual Euler totient for finite groups with representation implications
- Establishes a relation with K.S. Brown's problem

## Abstract

We extend the Euler's totient function (from arithmetic) to any irreducible subfactor planar algebra, using the Mobius function of its biprojection lattice, as Hall did for the finite groups. We prove that if it is nonzero then there is a minimal 2-box projection generating the identity biprojection. We explain a relation with a problem of K.S. Brown. As an application, we define the dual Euler totient of a finite group and we show that if it is nonzero then the group admits a faithful irreducible complex representation. We also get an analogous result at depth 2, involving the central biprojection lattice.