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arXiv · 1205.2742

The little desert? Some subfactors with index in the interval (5,3+\sqrt{5})

Scott Morrison·Emily Peters
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Abstract

Progress on classifying small index subfactors has revealed an almost empty landscape. In this paper we give some evidence that this desert continues up to index 3+\sqrt{5}. There are two known quantum-group subfactors with index in this interval, and we show that these subfactors are the only way to realize the corresponding principal graphs. One of these subfactors is 1-supertransitive, and we demonstrate that it is the only 1-supertransitive subfactor with index between 5 and 3+\sqrt{5}. Computer evidence shows that any other subfactor in this interval would need to have rank at least 38. We prove our uniqueness results by showing that there is a unique flat connection on each graph. The result on 1-supertransitive subfactors is proved by an argument using intermediate subfactors, running the `odometer' from the FusionAtlas` Mathematica package and paying careful attention to dimensions.

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Scott Morrison, Emily Peters. 2012-05-12. The little desert? Some subfactors with index in the interval (5,3+\sqrt{5}). https://doi.org/10.1142/s0129167x14500803

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