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.

Consider the definition of a module ${M}$ over the semidirect product ${S_n \ltimes A^{\otimes n}}$, for ${A}$ an associative algebra over ${\mathbb{C}}$. By definition, ${S_n }$ is imbedded in this algebra, so ${M}$ is an ${S_n}$-representation. In addition, we have maps ${y_{i,a}: M \rightarrow M}$ which define “multiplication by ${a}$ in the ${i}$th spot,” i.e. multiplication by ${(1 \otimes 1 \otimes \dots \otimes a \otimes \dots \otimes 1) \in S_n \ltimes A^{\otimes n}}$. This takes care of the generators. But we need to know what the relations are. First, by definition of the semidirect product, there has to be multiplicativity

$\displaystyle y_{i,a} \circ y_{i,b} = y_{i, ab} .$

In addition, assuming ${A}$ is unital, we should have ${y_{i, 1} = \mathrm{id}}$ for any ${i}$.

Next, the action of the ${y_i}$ must respect the permutations of ${S_n}$ because this is a semidirect product. The definition of a semidirect product means that we have to have ${\sigma y_{i,a} \sigma^{-1} = y_{\sigma(i), a}}$.

Finally, ${A^{\otimes n}}$ is a tensor product of algebras and we thus have a commutativity constraint: what goes on in the ${i}$-th spot and the ${j}$-th spot are independent and do not influence each other. In particular, ${[y_{i,a}, y_{j,b}] = 0}$ if ${j \neq i}$. For ${i=j}$, we have ${[y_{i,a}, y_{i,b}] = y_{i,[a,b]}}$. Oh, and obviously ${a \rightarrow y_{i,a}}$ has to be a linear map… I think this hits everything.

So far, we’ve written out in gory detail what it means to be a representation of ${S_n \ltimes A^{\otimes n}}$. But this isn’t very helpful yet for interpolation. The condition states that we have to have ${n}$ linear maps ${y_{i,a}: M \rightarrow M}$, and that makes no sense when ${n}$ is complex. Besides, the ${y_{i,a}}$ are just linear maps of vector spaces, not ${S_n}$-homomorphisms, and that also doesn’t make sense in complex rank.

However, there is a trick. We regard each family of maps ${y_{i,a}}$ for ${a}$ fixed as a map ${y_a: \mathfrak{h} \otimes M \rightarrow M}$. Then this is indeed a homomorphism of ${S_n}$-representations, by the condition ${\sigma y_{i,a} \sigma^{-1} = y_{\sigma(i), a}}$.

This means that the appropriate definition of the category ${\mathrm{Rep}(S_t \ltimes A^{\otimes t})}$ for ${t}$ complex should consist of objects ${M \in \mathrm{Rep}(S_t)}$ equipped with morphisms ${y_a: \mathfrak{h} \otimes M \rightarrow M}$ for ${a \in A}$ satisfying a few more conditions. We need to write these definitions in a category-friendly form.

The key observation, however, was already stated: a family of linear maps ${f_i: M \rightarrow M}$ satisfying ${\sigma f_i \sigma^{-1} =f_{\sigma(i)}}$ for all ${i}$ is the same thing as a ${S_n}$-morphism ${f:\mathfrak{h} \otimes M \rightarrow M}$. The composition of two such families of maps ${\{f_i\}, \{g_i\}}$ can be encapsulated by the map

$\displaystyle \mathfrak{h} \otimes M \rightarrow \mathfrak{h} \otimes \mathfrak{h} \otimes M \rightarrow \mathfrak{h} \otimes M \rightarrow M$

where the first map comes from the dual ${r}$ to the algebra bilinear form ${\mathfrak{h} \otimes \mathfrak{h} \rightarrow \mathfrak{h}}$ (which dual just sends ${e_i \rightarrow e_i \otimes e_i}$ for ${\{e_i\}}$ the standard basis for the usual representation). So the condition ${y_{i,a} \circ y_{i,b} = y_{i,ab}}$ becomes

$\displaystyle y_{a} \circ (1_{\mathfrak{h}} \otimes y_b) \circ (r \otimes 1_M) = y_{ab} : \mathfrak{h} \otimes M \rightarrow M.\ \ \ \ \ (1)$

This is in a sense that makes perfect sense in Deligne’s categories, because they too have a tensor structure and the map ${r}$ can be defined using relations (exercise left to the reader).

But, we have a few more conditions. The one that ${y_{i,1} = 1_M}$ for all ${i}$ is easy: it becomes

$\displaystyle y_1 = (s \otimes 1_M) \ \ \ \ \ (2)$

where ${s: \mathfrak{h} \rightarrow 1}$ is the morphism sending each ${e_i \rightarrow 1}$. The map ${s }$ is defined in complex rank (another exercise), so this condition is also category-friendly.

Next, we have linearity: the association ${a \rightarrow y_a}$ must be ${\mathbb{C}}$-linear. This one is obviously meaningful for Deligne’s categories, and hence carries over to complex rank.

Last, and hardest, is the condition on the commutators. I will simply state the condition; the patient reader can verify that it is a category-friendly version of the usual one.

$\displaystyle y_a \circ (1_{\mathfrak{h}} \otimes y_b) - y_b \circ (1_{\mathfrak{h}} \otimes y_a) \circ P_{1,2} = y_{[a,b]} \circ (\rho \otimes 1_{\mathfrak{h}} ) \colon \mathfrak{h} \otimes \mathfrak{h} \otimes M \rightarrow M. \ \ \ \ \ (3)$

Hence, Etingof made the following definition:

Definition 1 An object of ${\mathrm{Rep}(S_t \ltimes A^{\otimes t})}$ is an object ${M \in \mathrm{Rep}(S_t)}$ (Deligne’s category) together with a linear association of ${y_a: \mathfrak{h} \otimes M \rightarrow M}$ to each ${a \in A}$, satisfying the equations above (1), (2), (3). A morphism in this category is one commuting with the ${y}$-maps.

Granted, wreath products are not exactly exotic. There is a variant $\mathbf{H}_n$ of the semidirect product with the polynomial ring, called the degenerate affine Hecke algebra of type ${A}$ (dAHA), where the generators are ${\sigma \in S_n, y_i \ (1 \leq i \leq n)}$ satisfying the usual ${\sigma y_i \sigma^{-1} = y_{\sigma(i)}}$ and the unusual

$\displaystyle [y_i , y_j] = \frac{1}{4} \sum_{k \neq i,j} (kij) - (kji) .$

So this algebra ${\mathbf{H}_n}$ is kinda like the semidirect product of ${S_n}$ with the polynomial ring in ${n}$ variables, except the variables don’t exactly commute. Their commutator is not terrible, though; it’s in the group algebra.

The representation theory of this algebra is related to that of other (presumably more complicated?) Hecke algebras, which in turn are useful in the Langlands program and various other things. However, I don’t understand any of that; this was definitely an obstacle when I started my project (since the dAHA in complex rank was the first thing I started looking at last summer). What I can tell you is that Etingof found a way to incorporate even the messy relation above defining the Hecke algebra into complex rank. The idea is that the sum over three-cycles can be represented as a commutator of suitable Jucys-Murphy elements, and the definition Etingof made was:

Definition 2 An element of ${\mathrm{Rep}(\mathbf{H}_t)}$ is an element of ${\mathrm{Rep}(S_t)}$ together with a morphism ${y: \mathfrak{h} \otimes M \rightarrow M}$ satisfying$\displaystyle y \otimes (1 \otimes y) \otimes (1-P_{1,2} ) = (B_{\mathfrak{h}} \otimes 1_M) \circ [\Omega^{1,3}, \Omega^{2,3}]: \mathfrak{h} \otimes \mathfrak{h} \otimes M \rightarrow M.$

Morphisms are morphisms in ${\mathrm{Rep}(S_t)}$ commuting with the ${y}$-maps.

I should explain what the terms are. Here ${P_{1,2}}$ flips the two ${\mathfrak{h}}$‘s in ${\mathfrak{h} \otimes \mathfrak{h} \otimes M \rightarrow M}$ (this is asymmetric tensor category, recall); ${B_{\mathfrak{h}}: \mathfrak{h} \otimes \mathfrak{h} \rightarrow \mathbf{1}}$ is the morphism of self-duality, which makes sense in complex rank; ${\Omega^{1,3}, \Omega^{2,3}}$ are the Jucys-Murphy endomorphisms acting on the first and third (resp. second and third) factors. It is possible to check by some careful calculation that this messy equation actually does encapsulate the commutation relation above. Note that, as we saw yesterday, the Jucys-Murphy endomorphisms do make sense!

There are tons of other categories defined. For instance, there are degenerate affine Hecke algebras of type ${B}$ built out of the wreath products ${S_n \ltimes (\mathbb{Z}/2\mathbb{Z})^n}$; their categories of representations can be defined in complex rank using Knop’s categories of representations ${\mathrm{Rep}(S_t \ltimes (\mathbb{Z}/2\mathbb{Z})^t)}$. It is not my purpose to rewrite Etingof’s talk, though, but rather to provide an appetizer, so I refer the reader to it.

So, we’ve defined all these categories: now what to do with them? Well, recall that Deligne’s ${\mathrm{Rep}(S_t)}$ outside of the “small” set of the nonnegative integers exhibited somewhat similar behavior to the classical theory (a combinatorial Young-diagram based parametrization of simple objects, for instance). One expects that generically, these representation categories in complex rank will behave like the representation theory that has already been well-studied (namely, integral rank). One also expects that at algebraic parameters, funny things will happen–like the non-semisimple categories ${\mathrm{Rep}(S_t), t \in \mathbb{Z}_{\geq 0}}$. As far as I know, nobody has really found anything in the latter case (which is more difficult and significant); I only handle the former in my paper. Speaking of which, I’m finally going to get to that in my next post.