Galois groups and an obstruction to principal graphs of subfactors. (English) Zbl 1117.46041
Summary: 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 [Ann.Math.162, No.2, 581–642 (2005; Zbl 1125.16025)]. 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 \(7<n\leq 27\) by using the program GAP. We further utilize Mathematica to extend the statement to \(27<n\leq 55\). We conjecture that none of the graphs are principal graphs for all \(n>7\) and give 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 K.Ikeda [J. Math.Sci., Tokyo 5, No.2, 257–272 (1998; Zbl 0910.46044)].
MSC:
| 46L37 | Subfactors and their classification |
| 11R32 | Galois theory |
| 57M27 | Invariants of knots and \(3\)-manifolds (MSC2010) |
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