Galois groups and an obstruction to principal graphs of subfactors

Abstract

The Galois group of the minimal polymonal of a Jones index value gives a new type of obstruction to a principal graph, thanks to a recent result of P.Etingof, D.Nikshych, and V.Ostrik. We show that the sequence of the graphs given by Haagerup as candidates of principal graphs of subfactors, are not realized as principal graphs for 7<n <= 27 using GAP program. We further utilize Mathematica to extend the statement to 27 <n <= 55. We conjecture that none of the graphs are principal graphs for all n>7, and give an evidence using Mathematica for smaller graphs among them for n>55. The problem for the case n=7 remains open, however, it is highly likely that it would be realized as a principal graph, thanks to numerical computation by Ikeda.