[Updated, 6/12; various errors fixed]
I’ve just uploaded to arXiv my paper (submitted to J. of Algebra) “Categories parametrized by schemes and representation theory in complex rank,” an outgrowth of my RSI project started last summer, where I worked with Pavel Etingof and Dustin Clausen. I will devote this post to talking about some of the story surrounding it. In short, the project is about looking at this program of studying representation theory when the dimension is complex (admittedly nobody has ever seen a vector space of dimension ; I will explain this precisely below) in the simplest possible case. But the categories of interest in the project are built out of certain symmetric tensor categories that Deligne defined back in 2004, and I’ll talk a bit about those today. I could have just jumped straight into my paper, but I figured this would make things potentially more accessible, and would be more fun.
I also recommend looking at these posts of David Speyer and Noah Snyder, which talk about some of Deligne’s work as well (and which I learned a lot from). Also, cf. this talk of Pavel Etingof. The talk goes further (into the non-semisimple analogs of Deligne’s categories) that I will cover in a later post. Finally, the paper of Deligne is available here.
The whole story behind this starts with the representation theory of the classical groups—these are objects like , etc. And in particular, I’m going to zoom in on the symmetric group—or more precisely, the family of symmetric groups .
The symmetric group is a very complicated object (indeed, any finite group is a subgroup of a symmetric group, by Cayley’s theorem), but its representation theory has been understood for 100 years and has many interesting combinatorial facets.
In the modern language, we can package the entire representation theory of into a category (depending on the nonnegative integer ). This is a very interesting category for several reasons. The first, and most obvious, part of its structure is that it is a -linear abelian category.
More interestingly, it’s semisimple: every exact sequence splits. This is because the group algebra is semisimple, by Maschke’s theorem. In addition, it is a tensor category: we can define the tensor product of any two representations of a group in a natural way, and is no exception. It is even a symmetric tensor category because we have a nice isomorphism for any two representations .
Technically, all this works for any finite group. What’s special about the symmetric group is, for instance, the very nice way the simple objects of (i.e. irreducible representations) are parametrized. Namely, (as for every finite group) they are in bijection with the conjugacy classes of , but (unlike for other groups) we have an explicit map between such conjugacy classes and irreducible representations. Since each conjugacy class of corresponds to a partiton of (a well-known fact easily seen because any permutation can be written as a product of disjoint cycles),
The whole idea behind Deligne’s work is that, while there isn’t any such thing as a symmetric group on elements, there is nevertheless a category (or more generally for ) that has much of the same structure. Deligne constructed these categories via an interpolation procedure.