Jones Representations of Thompson’s Group F Arising from Temperley–Lieb–Jones Algebras

Abstract

Abstract Following a procedure due to 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 behavior 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 1st 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.