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…)