SF&PA: An example

Alright, let’s try to build a bi-oidal category with a good theory of duals. The dumbest possible way to do this is to start with a good monoidal category and put in meaningless labels by hand to make it bi-oidal. To be concrete, let’s consider the group S_3 and its category of unitary representations. The irreps of S_3 are the trivial irrep V^1, the sign irrep V^{-1}, and the standard rep V^2.

Let {}_A X_B = V^2 where A and B are meaningless labels. Note that {}_B X^*_A = V^2 since the standard rep is self-dual. Let’s see what bi-oidal category they tensor generate. Well, {}_A X_B \otimes_B {}_B X^*_A should mean take V^2 \otimes V^2 and think of that as an A-A object (again the labels here are totally meaningless). Note that V^2 \otimes V^2 \cong V^1 \oplus V^{-1} \oplus V^2. Continuing on in this fashion we get a bi-oidal category whose A-A, A-B, B-A, and B-B sectors are all just a copy of the representation theory of S_3, and all tensor products are just the tensor products as S_3 representations.

Now let’s try doing this same process with the dihedral group D_{2 \cdot 4} again starting with X being the two dimensional irrep. An easy computation shows that you end up with a slightly different picture. The A-A and B-B sectors consist of direct sums of the 1-dimensional representations, while the A-B and B-A sectors consist only of direct sums of the 2-dimensional representation. This is also a subfactor category.

You can go through the same process with your favorite group and your favorite representation and get a new subfactor category.

(Warning! This construction is not what’s called a group subfactor, which I’ll get to soon.)

4 thoughts on “SF&PA: An example

  1. In general here’s what you get from this construction. After replacing G with its image in Aut(V) we can assume that V is a faithful representation. Then the A-A and B-B sectors are the G-reps with trivial central character, the A-B sector is the G-reps with the same central character as V, and the B-A sector is the G-reps with the same central character as V*.

  2. I’ll just note, by central character, Noah means center of the group, NOT center of the group algebra (there’s only one representation with any given character of the whole center of the group algebra). I may be the only person in the world with a twisted enough perspective on the representation theory of finite groups to have been mixed up about that, but there it is.

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