[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…)