From the drawers of the museum

One of my amateur interests is paleontology. Paleontologists looking for new examples have two options: go out in the field and dig up a new example, or go looking through drawers of museums to find old examples that had been overlooked. In this blog post I want to give an interesting example of the latter kind of research being useful in mathematics. Namely in discussions with Zhengwei Liu, we realized that an old example of Ocneanu’s gives an answer to a question that was thought to be open.

One of the central problems in fusion categories is to determine to what extent fusion categories can be classified in terms of finite groups and quantum groups (perhaps combined in strange ways) or whether there are exceptional fusion categories which cannot be so classified. My money is on the latter, and in particular I think extended Haagerup gives an exotic fusion category. However, there are a number of examples which seem to involve finite groups, but where we don’t know how to classify them in terms of group theoretic data. For example, the Haagerup fusion category has a 3-fold symmetry and may be built from \mathbb{Z}/3\mathbb{Z} or S_3 (as suggested by Evans-Gannon). The simplest examples of these kind of “close to group” categories, are called “near-group categories” which have only one non-invertible object and have the fusion rules

X^2 \cong X^{\oplus n} + \sum_g g

for some group of invertible objects g. A result of Evans-Gannon (independently proved by Izumi in slightly more generality), says that outside of a reasonably well understood case (where n = \#G -1 and the category is described by group theoretic data), we have that n must be a multiple of \# G. There are the Tambara-Yamagami categories where n = 0, and many examples (E6, examples of Izumi, many examples of Evans-Gannon) where n = \#G

Here’s the question: Are there examples where n is larger than \# G?

It turns out the answer is yes! In fact the answer is given by the 0-graded part of the quantum subgroup E_9 of quantum SU(3) from Ocneanu’s tables here. I’ll explain why below.

The category of representations of a group has a restriction functor to the category of representations of any subgroup. This suggests a generalization of the notion of “subgroup” to an arbitrary tensor category. If C is a tensor category, then a “quantum subgroup” (of type I) is a tensor category D with a tensor functor F: C \rightarrow D which is dominant (every object in D is a summand of an object in the image of F). In particular, this makes D into a module category. (A simple module category which doesn’t come from a tensor functor is called a subgroup of type II, Ocneanu’s list includes both types.)

Ocneanu’s notation here is as follows. The quantum subgroups of type I are the ones with a starred vertex. The vertices of the graph are the simple objects in D, and the starred vertex is the trivial object. The category of representations of quantum SU(3) is \mathbf{Z}/3\mathbf{Z}-graded. Sometimes this grading descends to a grading on D, if it does then Ocneanu denotes the grading by coloring the vertices white, grey, and black. Note that the zero graded vertices (the white ones) form a tensor subcategory.

The edges of the graph give the fusion rules for tensoring with the fundamental representations of SU(3). That is the number of edges from A to B is the dimension of the hom spaces between \mathrm{Hom}(V \otimes A, B) where V is a fundamental representation. In fact,there are two fundamental representations, V and V^*, and it is possible to distinguish which edge is which by looking at the coloring of vertices since V adds 1 to the grading and V^* subtracts 1.

(The graph is also decorated with some additional information, that won’t be needed here. For example, the vertices are circled if the object is “dyslectic” and the subcategory of dyslectic objects is braided.)

For E9 there are four objects which are 0-graded. Three of dimension 1 (which we call 1, g, and g^2), and one of dimension 3 + 2 \sqrt{3} which we will call X. We can compute that X \oplus 1 \cong V \otimes V^*. Thus we can work out the rules for tensoring with X by counting paths of length 2 which go white-black-white, but subtracting 1 from the total count of paths from a vertex to itself. Using this we can see that X \otimes X \cong 1 \oplus g \oplus g^2 \oplus 6X, where this 6 appears as 1+1+1+2^2-1. Of course, 6 > 3.

If you know what a conformal inclusion is (I only sort of do), this example comes from the conformal inclusion of SU(3) at level 9 including into the exceptional group E6 at level 1.

Also, it turns out that for the special case when G is \mathbf{Z}/3\mathbf{Z}, a result of (then high school student) Hannah Larson, shows that n can’t be any larger than 6. This example shows that her result is sharp.

I think as the amount of mathematics grows, it will be increasingly important to find better ways to arrange old information in ways that make searching easier. In a perfect world, there should be a searchable database of fusion categories where one could just ask for all known examples of rank 4 fusion categories with 3 invertible objects and have this example returned. (In this case, the example would definitely be in the database because the paper itself is well-known, it just has a huge list of examples.)

(Finally, I’d like to note that this example will eventually be mentioned in the formal literature in a paper of Zhengwei’s.)

2 thoughts on “From the drawers of the museum

  1. Nice post — thanks.

    Perhaps the Knot Atlas would be a good model for a fusion category atlas.

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