SF&PA: How complicated are groups? March 19, 2008Posted by Noah Snyder in group theory, planar algebras, subfactors.
One more warmup post before I get to actual Subfactors. If you were asked to rank finite groups in order of how complicated they were, what measurement would you use?
There are three candidates that come to mind quickly, and a fourth which is a little subtler.
First you could use #G. This has some nice properties, for example there are only finitely many groups of a given size, so you could hope to enumerate all groups of size say less than 100. Representation theoretically we can define #G as the sum of the squares of the dimensions of the irreps. In Subfactor land we will be calling this measurement of complicatedness the “global index.”
However, as a representation theorist this description has some problems. You could have a small group that nonetheless had many representations. So let’s define the rank of a group to be the number of irreps. This also gives a measure of complicatedness of a group. It is a fun excercise (due originally to Landau) to prove that there are only finitely many groups of a given rank (hint: 1 is the sum of #[g]/#G where [g] is the size of a conjugacy class). For example:
- Rank 1: The trivial group
- Rank 2: Z/2
- Rank 3: Z/3 and S_3
In Subfactor land, we will still call the analogous notion the rank.
Finally, suppose you actually wanted to do computations in a group. You’ll quickly discover that it’s much easier to do computations in the symmetric group S_100 than it is in Monster group, even though the former has much larger size and much larger rank. The reason for this is that S_100 has a 99-dimensional faithful representation where you can do calculations, where for the Monster you’re stuck with a 196882-dimensional irreducible representation. So we could instead measure the complicatedness by the dimension of the smallest faithful irrep.
Now the simplest groups are the cyclic groups, the next are the dihedral groups together with the three platonic solid groups, etc. Notice that we don’t have finitely many examples at each level anymore.
In Subfactor land things will be slightly different, the analogy with groups is that a Subfactor is like a group with a fixed choice of faithful representation. The dimension of this irrep is (the square root of) the index. This is the usual measure of complicatedness for subfactors, which was a source of confusion for me for a long time since I was thinking internally about size (or global index).
One final comment about using the size of a faithful irrep as the definition of complicatedness: it works just as well for infinite groups so long as they have faithful finite dimensional representations.
Finally there is a measure for complicatedness in subfactors which is a bit less intuitive than the above three. It is a standard exercise in group representation theory that given a faithful representation, every other irrep appears in a sufficiently high tensor power. Exactly how high a tensor power do you need? That is to say, how hard is it to describe the other irreps in terms of the irrep you already have? In Subfactor land this tensor exponent is called the depth.
For example, if we take S_4 with the standard representation, it has size 24, rank 5, its smallest faithful irrep has dimension 3, and it has depth 3 (since you need to go to the tensor cube of the standard in order to find the sign rep).
These four are the main measurements used in subfactor theory, but for groups I can think of at least one more. Let’s say that a group is least complicated when the group ring is closest to a matrix ring. That is to say it’s simple when the largest irrep is really large (hence that matrix factor takes up most of the group ring). Perhaps surprisingly it’s possible to say some nice things about what groups are “simple” with respect to this measurement (shameless plug).