[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…)