This post is meant as a warm-up to my planned follow-up to David’s post. You don’t have to have read his post to understand this one, but there are a few technical details at the end where I’ll refer you to the end of his post. Most of what I learned here I learned from reading this expository paper by Ostrik which I read in preparation for some talks I gave my second year of grad school.
If you like to draw pictures, how do you think about the representation theory of groups? Well, you use an oriented strand for some basic or fundamental representation V of a group, you orient the strand the other way for the dual representation, you use disjoint union for tensor product. Now you can try to draw pictures for maps between tensor products say. Stacking these pictures is composition, and disjoint union is tensor product. This should be pretty familiar to you if you’ve read the archives for “this week in mathematical physics.”
Since we’re looking at the category of representations of a group we have a bonus bit of information: this tensor category is symmetric. There’s a canonical map which satisfies the relations of the symmetric group. In pictures this can be drawn using a crossing. (Warning: this is not a crossing in 3-dimensional space, you need to either think of your pictures as being in 4-dimensions or not embedded at all.)
Ok, so what is ? It should be a linear symmetric tensor category with a representation V that has no properties other than having dimension t. What does it mean to have dimension t in picture language? It means that a closed loop should have the value t. So here’s our proposed category :
- Objects are collections of oriented points on a line.
- Morphisms are linear combinations of oriented strands (unembedded or in dimension greater than 4 so that the crossings satisfy the relations of the symmetric group) whose boundaries match the objects that they’re mapping between.
- Composition is stacking of diagrams with the relation that a closed loop can be removed for a multiplicative factor of t.
- Tensor product is disjoint union.
Here’s a typical morphism in this category:
Ok, if t is an integer have we recovered the usual category of representations of ? Well certainly not, because the only objects are tensor products of V and its dual, we don’t have anything like Sym^2 V. This isn’t a big problem, since V is faithful, every representation occurs as a summand in some tensor power of it. Hence to recover all the reps we need only formally adjoin images of all projections. This is called taking the idempotent completion, or the Karoubi envelope, and David’s post explains it nicely.
The second problem is a bit more subtle. We also want to kill all “negligible morphisms” (this is another way of saying kill the radical as in David’s post). A negligible morphism is a morphism that when you close it off in any way (thereby getting an endomorphism of the trivial object) you always get zero. The cool thing about the collection of negligible morphisms is that its the unique maximal planar ideal! If you’re going to kill any morphism you have to kill off a negligible morphism (otherwise you’d kill its closing off which would kill a nonzero multiple of the empty diagram and hence kill everything), and you can kill of all the negligble morphisms if you want to. Now why would you want to? Well for one thing if you want your category to be semisimple (as we know the category of representations of a reductive group must be) then you’d better kill the negligible morphisms. So killing off the negligible morphisms means we’re not just looking at any n-dimensional representation of a group, we want to make sure that we’re looking at a representation of a reductive group.
This is a little subtle. Can anyone come up with a reason to study GL_t for t an integer but without killing off the negligibles? Might it be useful say in studying the representation theory of the group of upper-triangular matrices? Or the supergroup GL(m|n) with m-n=t? I’m not sure. Is there a better reason than semismplicity (which is not an obvious property of GL_n) to explain why you should kill off all neglibiles?
And now here’s the miracle, once you fix these two technical issues you really do recover GL_n. It should be possible to give some category theoretical proof of it. Clearly there’s a functor from this GL_n to the usual category of representations of GL_n. So if you could show that this category really is the category of representations of a group then you could use the universal property of GL_n as the largest group acting on an n dimensional vector space to show that this functor is an equivalence. This last step requires using some more heavy-duty results of Deligne that I’m not going to get into.
I’ll finish this off with a few exercises:
- What’s the relationship between the above and Schur-Weyl duality?
- What about the orthogonal group O_t? What’s the only new property that the standard representation of the orthogonal group has? How would you write that in terms of pictures.
- Do all the above for quantum gl_t. (Hint: HOMFLY!)
- Interpret the above in terms of the representation theory of GL_n for n really large. Which kinds of stable representations are you talking about? What’s the combinatorics for describing them? (The answer to this explains the right combinatorics for simple objects in GL_t for t generic.)