This post continues my series on representation theory in complex rank, begun here with a discussion of Deligne’s interpolation of the representation categories of the symmetric group, introduced in his 2004 paper.

Semisimplicity is the basic structure theorem for Deligne’s categories, and I would be extremely remiss in my discussion of representation theory in complex rank if I did not say something about it.

So, let’s review. In the first post, I explained and motivated the definition of Deligne’s categories {\mathrm{Rep}(S_t)}. Incidentally, Deligne did the same for the other classical groups, i.e. {GL_n, O_n, Sp_{2n}}, but I shall not discuss them. The categories {\mathrm{Rep}(S_t)}, are defined as the pseudo-abelian envelope of the {\mathbb{C}}-linear category generated by objects {\mathfrak{h}^{\otimes p}}, where the hom-spaces {\hom( \mathfrak{h}^{\otimes p}, \mathfrak{h}^{\otimes r})} are free on the equivalence relations on {\mathbf{ p+r}}, and composition is given by a combinatorial expression which is polynomial in the rank {t} (hence interpolable).

Now, we just have an abstract category with formal objects and morphisms corresponding in no obvious way to anything concrete. To prove it is semisimple, we cannot use therefore techniques such as those in the proof of Maschke’s theorem of Weyl’s complete reducibility theorem.

But we can do it by appealing to what I discussed in the second post of this series: by proving that the endomorphism rings are semisimple and the category is nonnilpotent. In fact, since direct products and factor rings of semisimple rings are semisimple, we only need to prove that the algebras {\hom_{\mathrm{Rep}(S_t)}(\mathfrak{h}^{\otimes p}, \mathfrak{h}^{\otimes p})} are semisimple (in addition to nonnilpotence). This endomorphism ring (depending on the size {p} and the rank {t}) is an important object, called the partition algebra, and you can look it up e.g. here. But I don’t know how to prove directly that the partition algebra is semisimple. So I will follow Deligne (and Knop) in the (inductive) proof (which will also imply semisimplicity of the partition algebra).

I will do this in two steps. First, I will use a little bit of combinatorics to show that when {t \notin \mathbb{Z}_{\geq 0}}, the category {\mathrm{Rep}(S_t)} is nonnilpotent. Next, I will use this to prove semisimplicity.



[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 {\pi}; 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.

1. Motivation

The whole story behind this starts with the representation theory of the classical groups—these are objects like {S_n, GL(n),  O(n)}, etc. And in particular, I’m going to zoom in on the symmetric group—or more precisely, the family of symmetric groups {S_n, n \in \mathbb{Z}_{\geq 0}}.

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 {S_n} into a category {\mathrm{Rep}^{\mathrm{ord}}(S_n)} (depending on the nonnegative integer {n}). This is a very interesting category for several reasons. The first, and most obvious, part of its structure is that it is a {\mathbb{C}}-linear abelian category.

More interestingly, it’s semisimple: every exact sequence splits. This is because the group algebra {\mathbb{C}[S_n]} 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 {S_n} is no exception. It is even a symmetric tensor category because we have a nice isomorphism {X \otimes Y \rightarrow Y \otimes X} for any two representations {X,Y}.

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 {\mathrm{Rep}^{\mathrm{ord}}(S_n)} (i.e. irreducible representations) are parametrized. Namely, (as for every finite group) they are in bijection with the conjugacy classes of {S_n}, but (unlike for other groups) we have an explicit map between such conjugacy classes and irreducible representations. Since each conjugacy class of {S_n} corresponds to a partiton of {n} (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 {\pi} elements, there is nevertheless a category {\mathrm{Rep}(S_\pi)} (or more generally {\mathrm{Rep}(S_t)} for {t \in  \mathbb{C}}) that has much of the same structure. Deligne constructed these categories via an interpolation procedure.

2. Interpolation