This post continues my series on representation theory in complex rank, begun here with a discussion of Deligne’s interpolation of the representation categories of the symmetric group, introduced in his 2004 paper.

Semisimplicity is the basic structure theorem for Deligne’s categories, and I would be extremely remiss in my discussion of representation theory in complex rank if I did not say something about it.

So, let’s review. In the first post, I explained and motivated the definition of Deligne’s categories ${\mathrm{Rep}(S_t)}$. Incidentally, Deligne did the same for the other classical groups, i.e. ${GL_n, O_n, Sp_{2n}}$, but I shall not discuss them. The categories ${\mathrm{Rep}(S_t)}$, are defined as the pseudo-abelian envelope of the ${\mathbb{C}}$-linear category generated by objects ${\mathfrak{h}^{\otimes p}}$, where the hom-spaces ${\hom( \mathfrak{h}^{\otimes p}, \mathfrak{h}^{\otimes r})}$ are free on the equivalence relations on ${\mathbf{ p+r}}$, and composition is given by a combinatorial expression which is polynomial in the rank ${t}$ (hence interpolable).

Now, we just have an abstract category with formal objects and morphisms corresponding in no obvious way to anything concrete. To prove it is semisimple, we cannot use therefore techniques such as those in the proof of Maschke’s theorem of Weyl’s complete reducibility theorem.

But we can do it by appealing to what I discussed in the second post of this series: by proving that the endomorphism rings are semisimple and the category is nonnilpotent. In fact, since direct products and factor rings of semisimple rings are semisimple, we only need to prove that the algebras ${\hom_{\mathrm{Rep}(S_t)}(\mathfrak{h}^{\otimes p}, \mathfrak{h}^{\otimes p})}$ are semisimple (in addition to nonnilpotence). This endomorphism ring (depending on the size ${p}$ and the rank ${t}$) is an important object, called the partition algebra, and you can look it up e.g. here. But I don’t know how to prove directly that the partition algebra is semisimple. So I will follow Deligne (and Knop) in the (inductive) proof (which will also imply semisimplicity of the partition algebra).

I will do this in two steps. First, I will use a little bit of combinatorics to show that when ${t \notin \mathbb{Z}_{\geq 0}}$, the category ${\mathrm{Rep}(S_t)}$ is nonnilpotent. Next, I will use this to prove semisimplicity.

${\mathfrak{sl}_2}$ is a special Lie algebra, mentioned in my previous post briefly. It is the set of 2-by-2 matrices over ${\mathbb{C}}$ of trace zero, with the Lie bracket defined by:

$\displaystyle [A,B] = AB - BA.$

The representation theory of ${\mathfrak{sl}_2}$ is important for several reasons.

1. It’s elegant.
2. It introduces important ideas that generalize to the setting of semisimple Lie algebras.
3. Knowing the theory for ${\mathfrak{sl}_2}$ is useful in the proofs of the general theory, as it is often used as a tool there.

In this way, ${\mathfrak{sl}_2}$ is an ideal example. Thus, I am posting this partially to help myself learn about Lie algebras.