[This post, a continuation of the series on representation theory in complex rank, discusses the irreducibles in Deligne’s category \mathrm{Rep}(S_t) for t \notin \mathbb{Z}_{\geq 0} and what one can do with them.]

OK, so we now know that Deligne’s categories {\mathrm{Rep}(S_t)} are semisimple when {t \notin  \mathbb{Z}_{\geq 0}}. But, this is a paradox. Deligne’s categories, a family of categories constructed to interpolate the semisimple categories of representations of {S_n, n \in \mathbb{Z}_{\geq  0}} are semisimple precisely at the complement of the nonnegative integers!

The problem is, when {t \in \mathbb{Z}_{\geq 0}}, {\mathrm{Rep}(S_t)} is not equivalent to the ordinary category {\mathrm{Rep}^{\mathrm{ord}}(S_t)}. 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 {\mathrm{Rep}(S_t)} (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 {\mathrm{Rep}(S_t), t \notin \mathbb{Z}_{\geq 0}}, look like. We know what the simple {S_n}-representations are; they are the Specht modules, parametrized by the Young diagrams of size {n}. It turns out that the simple objects in {\mathrm{Rep}(S_t)} 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 {\mathrm{Rep}^{\mathrm{ord}}(S_n), n \in \mathbb{Z}_{\geq  0}}. Namely, we let {\mathrm{Rep}^{\mathrm{ord}}(S_n)^{(N)}} denote the category generated by {\mathfrak{h}^{\otimes p}, p \leq  N} for {\mathfrak{h}} the regular representation. When {N} is large enough, this becomes the full category, so we will always pretend that {n} is really really large relative to {N} (which is kinda ironic when you think about the notation…).

Anyhow, we want to look at the simple objects in {\mathrm{Rep}^{\mathrm{ord}}(S_n)^{(N)}}. Well, these are going to have to correspond to some Young diagrmas of size {n}, but the question is which ones?

I claim that the Young diagrams that arise are precisely those where the rows below the top have {\leq N} boxes.

In particular, as {n} gets large, the top row must get really long, but the number of simple objects stays bounded. (more…)

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.  (more…)