[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.

To prove this I shall use the Pieri rule in the representation theory of the symmetric group. Basically, we know that ${\mathfrak{h}}$ decomposes into irreducibles as ${\mathbf{1} \oplus \mathfrak{h}_0}$, where ${\mathbf{1}}$ corresponds to the Young diagram

and ${\mathfrak{h}_0}$ to the Young diagram

I’ve shown it in the case ${n=15}$, but the pattern should be clear: ${\mathbf{1}}$ has nothing below the top row, and ${\mathfrak{h}_0}$ only one square. In this way, it is easy to see that our result about the irreducibles in ${\mathrm{Rep}(S_n)^{(N)}}$ is valid for ${N=0,1}$.

Now, in proving the boldface claim, we are going to use some kind of inductive argument here, because the simple objects in ${\mathrm{Rep}(S_t)^{(N)}}$ outside of ${\mathrm{Rep}(S_t)^{(N-1)}}$ are precisely the factors of the simple objects in ${\mathrm{Rep}(S_t)^{(N-1)}}$ tensored with ${\mathfrak{h}}$. We can even replace ${\mathfrak{h}}$ with ${\mathfrak{h}_0}$, since ${\mathbf{1}}$ is the unital object.

We now quote the Pieri rule.

Theorem 1 Let ${X_{\lambda}}$ be the irreducible representation of ${S_n}$ corresponding to the Young diagram ${\lambda}$. Then$\displaystyle \mathfrak{h}_0 \otimes X_{\lambda} = \bigoplus X_{\mu} + C_{\lambda} X_{\lambda}.$

Here ${\mu}$ ranges over the set of Young diagrams that can be obtained by moving a corner cell, and ${C_{\lambda}}$ is the number of corner cells in ${\lambda}$.

Suppose the claim proved for ${\mathrm{Rep}(S_n)^{(N-1)}}$. Now any Young diagram ${\lambda}$ with ${\leq N}$ boxes below the top row can arise from a Young diagram ${\lambda'}$ with ${< N}$ boxes below the top row by a corner transformation. By induction, ${\lambda' \in \mathrm{Rep}(S_n)^{(N-1)}}$, whence ${\mathfrak{h}_0 \otimes X_{\lambda'} \in \mathrm{Rep}(S_n)^{(N)}}$, and ${X_{\lambda}}$ is a direct factor of ${\mathfrak{h}_0 \otimes X_{\lambda'} }$ by the Pieri rule. So, the proof now proceeds by induction.

So, the claim’s proved. Note that the claim is for ${n}$ large and ${N}$ fixed, and equivalently it gives a bijection between the simples in ${\mathrm{Rep}^{\mathrm{ord}}(S_n)^{(N)}}$ and the partitions of size ${\leq N}$ (by throwing away the top row). One thus expects that for ${t \notin \mathbb{Z}_{\geq 0}}$, the simple objects of ${\mathrm{Rep}(S_t)^{(N)}}$ are parametrized by Young diagrams of size at most ${N}$. In fact, it is a general phenomenon that “generically,” representation theory in complex rank looks like classical representation theory.

More generally, the simple objects in ${\mathrm{Rep}(S_t)}$ are parametrized by all Young diagrams, where a Young diagram of size ${N}$ can be heuristically viewed as a “Young diagram” of “size ${t}$” by adding a “top row” of “length” ${t-N}$.

Why is this ideology so useful? Well, here is an example. There is an important central element in the group algebra of the symmetric group ${\mathbb{C}[S_n]}$ called the Jucys-Murphy element ${\Omega}$; it is the sum over the two-cycles. Now, the Jucys-Murphy element by itself does not make any sense in Deligne’s categories, because there is no ${S_t}$ for ${t \notin \mathbb{Z}_{\geq 0}}$. However, there is still a corresponding endomorphism of the identity functor.

(Incidentally, it is a useful general bit of abstract nonsense that the endomorphisms of the identity functor on the category ${Mod(A)}$ of modules over a ring ${A}$ is isomorphic to the center of ${A}$.)

How are we going to get this endomorphism of the identity functor? Well, ${\mathrm{Rep}(S_t)}$ is semisimple when ${t \notin \mathbb{Z}_{\geq 0}}$, so we need only prescribe the maps ${X_\lambda \rightarrow X_\lambda}$ for ${\lambda}$ a Young diagram and ${X_\lambda}$ the associated simple object.

Now, consider a standard Young diagram ${\mu}$:

Suppose all but the top row is fixed, where the top row is allowed to vary in length and get longer and longer. Then, instead of one ${\mu}$, we have a family of Young diagrams ${\mu(n)}$ of size ${n}$. Consider the associated simple objects ${V_{\mu(n)} \in \mathrm{Rep}^{\mathrm{ord}}(S_n)}$. It can be shown, using the Frobenius character formula, that the action of ${\Omega}$ on ${V_{\mu(n)}}$ is a scalar polynomial in ${n}$. It is called the content.

In this way, one can define the content of a “Young diagram” where the top line has nonintegral length, just by interpolation.

So, now let’s go back to the partition ${\lambda}$ and the simple object ${X_{\lambda} \in \mathrm{Rep}(S_t)}$. The way to define ${\Omega}$ as a morphism ${X_{\lambda} \rightarrow X_{\lambda}}$ is to add a very long line atop ${\lambda}$ of “size” ${t - |\lambda|}$, and evaluate the content of this diagram(!).

(There is a fair bit of laughter in the middle of Etingof’s talk when he pulls out a transparency with a Young diagram with a comically large top row and normal-sized everything else.)

One disadvantage of this method is that you don’t get the Jucys-Murphy endomorphism for ${\mathrm{Rep}(S_t), t \in \mathbb{Z}_{\geq 0}}$. There is another way one can describe it explicitly in terms of the relation symbols used in the definition of Deligne’s categories, and this is actually the way that I find more useful in my paper. But this is fun.

2. Proof

Enough heuristic and ideological motivation. I am now finally going to prove the parametrization of the simple objects in ${\mathrm{Rep}(S_t)}$, and though it is a formality, I will enclose the theorem in its fancy box:

Theorem 2 For ${t \notin \mathbb{Z}_{\geq 0}}$, the simple objects in ${\mathrm{Rep}(S_t)^{(N)}}$ are indexed by the partitions of size ${\leq N}$. The simple objets in ${\mathrm{Rep}(S_t)}$ are indexed by partitions of arbitrary size.

This theorem is proved by induction on ${N}$ (it’s trivial when ${N=-1}$), and it uses a lot of what I set up yesterday. I will briefly review what I need; for the most part, you can refer to the post. So, let’s assume the theorem is true for ${N-1}$. Recall that we have a splitting

$\displaystyle \mathfrak{h}^{\otimes N} = V_N \oplus W_N$

where ${\hom(V_N, X) = \hom(X, V_N) = 0}$ for ${X \in \mathrm{Rep}(S_t)^{(N-1)}}$. We proved that

$\displaystyle End(V_N) \simeq \mathbb{C}[S_N].$

And now, we will use the representation theory of ${S_N}$ and some general nonsense to construct the irreducible objects in ${\mathrm{Rep}(S_t)^{(N)}}$.

First, we have central idempotents ${e_{\lambda} \in \mathbb{C}[S_N]}$ for ${\lambda}$ a Young diagram of size ${N}$ (indeed, the central idempotents are in bijection with the irreducible representations) and as a result, we get a splitting ${V_N = \bigoplus_{\lambda} Y_\lambda }$.

We have ${\hom(Y_{\lambda}, Y_{\lambda'}) = 0}$ if ${\lambda \neq \lambda'}$ because then the idempotents are both central and orthogonal. In addition, the endomorphism ring of each ${Y_{\lambda}}$ is a matrix algebra (in view of the Wedderburn structure theorem). However, ${Y_{\lambda}}$ is generally not going to be simple (it will be iff the corresponding irreducible representation has degree 1, by basic facts about regular representations).

Nevertheless, the following lemma implies that ${Y_{\lambda}}$ is isotypic, i.e. isomorphic to a direct sum of copies of the same simple object. It follows thus that the simple objects in ${\mathrm{Rep}(S_t)^{(N)}}$ not belonging to ${\mathrm{Rep}(S_t)^{(N-1)}}$ are indexed by the Young diagrams of size ${N}$, and so the lemma completes the proof the theorem.

Lemma 3 Suppose ${\mathcal{C}}$ is a semisimple ${\mathbb{C}}$-linear category and ${X \in \mathcal{C}}$ satisfies ${End(X) \simeq M_n(\mathbb{C})}$. Then ${X}$ is a direct sum of ${n}$ copies of a simple object.

Indeed, we have idempotents ${e_1, \dots, e_n}$ in the matrix algebra ${M_n(\mathbb{C})}$ which are just diagonal matrices with a 1 in the ${i}$th place for ${e_i}$ and zero everywhere else! There are corresponding simple objects ${X_1, \dots, X_n}$. Then ${\hom(X_i, X_j) \simeq e_i M_n(\mathbb{C}) e_j }$ is one-dimensional for all ${i,j}$, so the ${X_i}$ are isomorphic. It follows that ${End(\bigoplus X_i) = M_n(\mathbb{C})}$, so we have ${X = \bigoplus X_i}$ in fact.

Deligne is a little more environmentally friendly than I am when he discusses this material; he actually combines the proofs of semisimplicity and classification into one rather terse section.

Next time, I’ll stop discussing Deligne’s paper, and focus on the material in Etingof’s talk (namely, various non-semisimple generalizations of Deligne’s categories). I’m drawing this material not only from that talk and various papers, but also on various discussions I’ve had with him since last summer. Some of it seems to be unpublished “folklore,” and I’ll do my best to put this folklore in a form that might be useful to others.