Modular data for the extended Haagerup subfactor

TL;DR

This paper computes the modular data for the center of the extended Haagerup subfactor, revealing how representation theory aids in understanding its structure despite incomplete associativity data.

Contribution

It provides the first explicit computation of the modular data for this subfactor's center using combinatorial and representation theory methods.

Findings

Computed the S and T matrices for the extended Haagerup subfactor

Demonstrated the use of SL(2, Z) representation theory in modular data analysis

Explored character vectors associated with the modular data

Abstract

We compute the modular data (that is, the $S$ and $T$ matrices) for the centre of the extended Haagerup subfactor. The full structure (i.e. the associativity data, also known as 6-$j$ symbols or $F$ matrices) still appears to be inaccessible. Nevertheless, starting with just the number of simple objects and their dimensions (obtained by a combinatorial argument in arXiv:1404.3955) we find that it is surprisingly easy to leverage knowledge of the representation theory of $SL (2, \mathbb Z)$ into a complete description of the modular data. We also investigate the possible character vectors associated with this modular data.