# Jones representations of Thompson's group $F$ arising from Temperley-Lieb-Jones algebras

## TL;DR

This paper constructs a family of unitary representations of Thompson's group F using Temperley-Lieb-Jones algebras, analyzes their properties, and classifies their equivalence classes and stabilizer subgroups based on subfactor indices.

## Contribution

It introduces a new 3-parameter family of representations of F derived from Temperley-Lieb-Jones algebras and classifies their equivalence and stabilizer subgroups.

## Key findings

- Representations do not contain finite type components.
- Identified three inequivalent classes of representations.
- Stabilizer subgroups are trivial for large subfactor indices.

## Abstract

Following a procedure due to V. Jones, using suitably normalized elements in a Temperley-Lieb-Jones (planar) algebra we introduce a 3-parametric family of unitary representations of the Thompson's group $F$ equipped with canonical (vacuum) vectors and study some of their properties. In particular, we discuss the behaviour at infinity of their matrix coefficients, thus showing that these representations do not contain any finite type component. We then focus on a particular representation known to be quasi-regular and irreducible and show that it is inequivalent to itself once composed with a classical automorphism of F. This allows us to distinguish three equivalence classes in our family. Finally, we investigate a family of stabilizer subgroups of $F$ indexed by subfactor Jones indices that are described in terms of the chromatic polynomial. In contrast to the first non-trivial index value for which the corresponding subgroup is isomorphic to the Brown-Thompson's group $F_3$, we show that when the index is large enough this subgroup is always trivial.