We have now discussed some of the basic properties of Deligne’s categories {\mathrm{Rep}(S_t)}, and some of the rich structure that they have. It turns out, as I have already mentioned, that Deligne did the same for representation categories of the other classical groups.

Knop described how to do it for the wreath products, obtaining categories {\mathrm{Rep}(S_t \ltimes G^t)} for {t \in \mathbb{C}}; here the central object is the “standard representation” {\mathfrak{h}_G} of {G}-invariant functions {G \rightarrow  \mathbb{C}oprod_n G}, which has a natural action of {S_n \rtimes G^n}. The representation {\mathfrak{h}_G} is faithful, and again one uses its tensor powers and a combinatorial parametrization of its morphisms to interpolate. For the details in much more generality, see Knop’s paper; he actually constructs tensor categories via the calculus of relations out of arbitrary “regular categories.” (My paper has a brief exposition of how things play out in the special case of {\mathrm{Rep}(S_t \ltimes G^t)}.) These categories, like Deligne’s, are semisimple symmetric tensor categories.

It turns out, however, that many families of algebraic objects of interest in representation theory depend on a parameter {n \in  \mathbb{Z}_{\geq 0}}, and are built out of the corresponding (i.e., depending on {n}) classical groups (i.e. symmetric, orthogonal, etc.). One example is the family of algebras {S_n \ltimes A^{\otimes n}} for {A} an associative algebra. This is a rather simple one; a more complicated one is given by the family of Hecke algebras. The additional relations and generators corresponding to the part of these objects not in the classical groups can, however, often be stated in a uniform, categorical manner independent of {n}.

Using this, Etingof proposed a program of studying the representation categories of these objects in complex rank, which he constructed out of Deligne’s categories. I will briefly explain what this is all about. Consider the example of the family of semidirect product algebras; it’s simpler than what Etingof focuses on, but I’d be horrendously unqualified to really say anything about any of them. (more…)

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I was initially going to talk about why Deligne’s categories of representations of the symmetric group on a nonintegral number of elements are semisimple generically. This is a rather difficult result, and takes quite a bit of preparation in his paper.  However, I got sidetracked. Instead, I will devote this post to a general discussion of semisimple categories.  According to the material here, it follows that in order to show that Deligne’s categories are semisimple, one has to show that the so-called “partition algebra” is a semisimple ring.


1. Review of semisimple categories

Before we specialize to the case of Deligne’s categories, it may help to go through a little abstract nonsense. Suppose {\mathcal{C}} is a semisimple category. This means that {\mathcal{C}} is abelian, and each object in {\mathcal{C}} is a direct sum of simple objects, where simple means that there is no proper subobject. So for instance, the {A}-modules for {A} a semisimple algebra form a semisimple category. The finite-dimensional representations of a semisimple Lie algebra form a semisimple category (though the finite-dimensional condition is necessary; the enveloping algebra is not a semisimple algebra generally).

Now, I want to look at the hom-spaces in a semisimple category. But first, in the next lemma, there is no need to have the semisimplicity asumption, so I drop that.

Remember Schur’s lemma—that lemma in group representation theory, that any morphism between irreducible representations over {\mathbb{C}} is a scalar? The proof of it in different textbooks tends to vary between nonintuitive and clean (depending on the extent of the allegiance of said textbook to category theory). Because when thought of categorically, I claim that it is trivial.

Lemma 1 (Schur, categorical version) Let {X} be a simple object in a {\mathbb{C}}-linear abelian category with finite-dimensional hom-spaces. Then {\hom(X,X) \simeq  \mathbb{C}}. Also, {\hom(X,Y) =0} if {X,Y} are simple and nonisomorphic.

So, let’s prove this. We will first prove that any morphism between simple objects {X,Y} is an isomorphism or zero. If one were not zero, it would have either a nontrivial kernel or cokernel. And this would mean either that {X} had a nontrivial subobject or {Y} a nontrivial subobject—two things that can’t happen for simple {X,Y}.

It is now clear that {\hom(X,Y) = 0} when {X,Y} are nonisomorphic, because a nontrivial morphism would be an isomorphism by the above.

Well, then {\hom(X,X)} is a ring where every nonzero element is invertible—that is to say, a division algebra. It is also finite-dimensional over {\mathbb{C}}by the assumption on the hom-spaces. But every f.d. division algebra over {\mathbb{C}} is {\mathbb{C}} itself; indeed, if {\alpha \notin \mathbb{C}} belonged to such a division algebra, then {\mathbb{C}(\alpha)} would be a finite extension field (yes, commutative—{\alpha} commutes with itself!) and this cannot happen since {\mathbb{C}} is algebraically closed.

In particular, {\hom(X,X) = \mathbb{C}}. This proves Schur’s lemma. Not entirely trivial, but at least swift.

So that’s done. I claim then that, in a semisimple category {\mathcal{C}}, the hom spaces {\hom(X,X)} is ring-isomorphic to a product of matrix algebras over {\mathbb{C}}. This is now straightforward: decompose {X} as a sum of simple objects {S_1 \oplus S_2 \oplus \dots \oplus S_k}. Partition {S_1, \dots, S_k} into equivalence classes based on isomorphism and take the sums {T_j, 1 \leq j  \leq l} of the {S_i} in each equivalence class. Each {T_j} has hom-spaces isomorphic to a matrix algebra, so the claim is clear.

In particular, the hom-rings of a semisimple category are—surprise, surprise—semisimple algebras!

2. What if the hom-spaces are semisimple?

The 45-million-dollar question now arises whether the opposite might be true. In fact, I think it is, with certain hypotheses: this isn’t really about Deligne’s paper anymore, but it’s something that I learned from Friedrich Knop’s very interesting paper “Tensor envelopes of regular categories.” Knop actually generalizes Deligne’s construction and axiomatizes it to constructing large classes of interesting tensor categories (such as representation categories of wreath products {S_t \ltimes  G^{t}} for {G} finite and {t} complex. I may talk more about Knop’s paper later, but right now I am just using it as a source of some fun abstract nonsense.

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