SF&PA: An example March 25, 2008Posted by Noah Snyder in group theory, planar algebras, subfactors.
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 , the sign irrep , and the standard rep .
Let where A and B are meaningless labels. Note that since the standard rep is self-dual. Let’s see what bi-oidal category they tensor generate. Well, should mean take and think of that as an A-A object (again the labels here are totally meaningless). Note that 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 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.)