The ideology of a long top row: Or, why you shouldn’t be surprised about the irreducibles in Rep(S_t)

[This post, a continuation of the series on representation theory in complex rank, discusses the irreducibles in Deligne’s category for and what one can do with them.]

OK, so we now know that Deligne’s categories are semisimple when . But, this is a paradox. Deligne’s categories, a family of categories constructed to interpolate the semisimple categories of representations of are semisimple precisely at the complement of the nonnegative integers!

The problem is, when , is not equivalent to the ordinary category . The problem is that not all relations correspond to actual morphisms. Deligne in fact shows that the ordinary category can be obtained as a quotient of his (via the tensor ideal of “neglligible morphisms”) but this isn’t really important for the story I’m telling.

1. Motivation and remarks

Today, I want to talk about what the simple objects in , look like. We know what the simple -representations are; they are the Specht modules, parametrized by the Young diagrams of size . It turns out that the simple objects in are parametrized by the Young diagrams of arbitrary size. There is an interesting way of thinking about this that Etingof explains in his talk, and which I will try to motivate here now.

OK. So, just as we defined a filtration on Deligne’s categories yesterday, let’s define a filtration on the ordinary representation categories . Namely, we let denote the category generated by for the regular representation. When is large enough, this becomes the full category, so we will always pretend that is really really large relative to (which is kinda ironic when you think about the notation…).

Anyhow, we want to look at the simple objects in . Well, these are going to have to correspond to some Young diagrmas of size , but the question is which ones?

I claim that the Young diagrams that arise are precisely those where the rows below the top have boxes.

In particular, as gets large, the top row must get really long, but the number of simple objects stays bounded.

 

Deligne’s category Rep(S_t) for t not necessarily an integer

[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.

1. Motivation

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.

2. Interpolation

 

A quick lemma on group representations

So, since I’ll be talking about the symmetric group a bit, and since I still don’t have enough time for a deep post on it, I’ll take the opportunity to cover a quick and relevant lemma in group representation theory (referring as usual to the past blog post as background).

A faithful representation of a finite group is one where different elements of induce different linear transformations, i.e. is injective. The result is

Lemma 1 If is a faithful representation of , then every simple representation of occurs as a direct summand in some tensor power .  

 

Representations of the symmetric group

I’ve now decided on future plans for my posts. I’m going to alternate between number theory posts and posts on other subjects, since I lack the focus have too many interests to want to spend all my blogging time on one area.

For today, I’m going to take a break from number theory and go back to representation theory a bit, specifically the symmetric group. I’m posting about it because I don’t understand it as well as I would like. Of course, there are numerous other sources out there—see for instance these lecture notes, Fulton and Harris’s textbook, Sagan’s textbook, etc.  Qiaochu Yuan has been posting on symmetric functions and may be heading for this area too, though if he does I’ll try to avoid overlapping with him; I think we have different aims anyway, so this should not be hard.