Spectral Measures and Generating Series for Nimrep Graphs in Subfactor Theory II: SU(3)

TL;DR

This paper completes the calculation of spectral measures for SU(3) nimrep graphs in subfactor theory, revealing their distribution on special subsets of the discoid and torus, extending previous work on SU(2) and SU(3) graphs.

Contribution

It provides the full spectral measure computations for SU(3) nimrep graphs, including ADE and McKay graphs, advancing the understanding of their spectral properties.

Findings

Spectral measures for SU(3) graphs are supported on specific subsets of the discoid and torus.

The theory of nimreps enables precise computation of these spectral measures.

Results extend previous spectral measure analyses from SU(2) to SU(3).

Abstract

We complete the computation of spectral measures for SU(3) nimrep graphs arising in subfactor theory, namely the SU(3) ADE graphs associated with SU(3) modular invariants and the McKay graphs of finite subgroups of SU(3). For the SU(2) graphs the spectral measures distill onto very special subsets of the semicircle/circle, whilst for the SU(3) graphs the spectral measures distill onto very special subsets of the discoid/torus. The theory of nimreps allows us to compute these measures precisely. We have previously determined spectral measures for some nimrep graphs arising in subfactor theory, particularly those associated with all SU(2) modular invariants, all subgroups of SU(2), the torus, SU(3), and some SU(3) graphs.