We have now discussed some of the basic properties of Deligne’s categories , 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 for ; here the central object is the “standard representation” of -invariant functions , which has a natural action of . The representation 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 .) 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 , and are built out of the corresponding (i.e., depending on ) classical groups (i.e. symmetric, orthogonal, etc.). One example is the family of algebras for 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 .

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 over the semidirect product , for an associative algebra over . By definition, is imbedded in this algebra, so is an -representation. In addition, we have maps which define “multiplication by in the th spot,” i.e. multiplication by . 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

In addition, assuming is unital, we should have for any .

Next, the action of the must respect the permutations of because this is a semidirect product. The definition of a semidirect product means that we have to have .

Finally, is a tensor product of algebras and we thus have a commutativity constraint: what goes on in the -th spot and the -th spot are independent and do not influence each other. In particular, if . For , we have . Oh, and obviously 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 . But this isn’t very helpful yet for interpolation. The condition states that we have to have linear maps , and that makes no sense when is complex. Besides, the are just linear maps of vector spaces, not -homomorphisms, and that also doesn’t make sense in complex rank.

However, there is a trick. We regard each family of maps for fixed as a map . Then this is indeed a homomorphism of -representations, by the condition .

This means that the appropriate definition of the category for complex should consist of objects equipped with morphisms for 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 satisfying for all is the same thing as a -morphism . The composition of two such families of maps can be encapsulated by the map

where the first map comes from the dual to the algebra bilinear form (which dual just sends for the standard basis for the usual representation). So the condition becomes

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

But, we have a few more conditions. The one that for all is easy: it becomes

where is the morphism sending each . The map is defined in complex rank (another exercise), so this condition is also category-friendly.

Next, we have linearity: the association must be -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.

Hence, Etingof made the following definition:

Definition 1An object of is an object (Deligne’s category) together with a linear association of to each , satisfying the equations above (1), (2), (3). A morphism in this category is one commuting with the -maps.

Granted, wreath products are not exactly exotic. There is a variant of the semidirect product with the polynomial ring, called the degenerate affine Hecke algebra of type (dAHA), where the generators are satisfying the usual and the unusual

So this algebra is kinda like the semidirect product of with the polynomial ring in 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 2An element of is an element of together with a morphism satisfying

Morphisms are morphisms in commuting with the -maps.

I should explain what the terms are. Here flips the two ‘s in (this is asymmetric tensor category, recall); is the morphism of self-duality, which makes sense in complex rank; 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 built out of the wreath products ; their categories of representations can be defined in complex rank using Knop’s categories of representations . 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 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 . 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.

June 20, 2010 at 1:42 pm

[…] Categories parametrized by schemes and representation theory in complex rank June 20, 2010 tags: arXiv, Chevalley's theorem, constructible sets, Pavel Etingof, Pierre Deligne, representation theory in complex rank by Akhil Mathew I’d now like to discuss my paper “Categories parametrized by schemes and representation theory in complex rank.” My RSI project was rather-open ended: to investigate the categories of representation theory in complex rank. Pavel Etingof told me that it would be expected that they would behave similar in some ways to the integral case (at least if “there was justice in the world”). For instance, we know that Deligne’s has a comibnatorial parametrization of simple objects similar to the classical case. However, as I discovered when I got there, I don’t actually know representation theory. I had looked through some material on finite groups, and knew (in outline, not usually proofs) the basic facts about the symmetric group. I certainly didn’t know anything about Hecke algebras (the literature of which seems rather inaccessible to beginners), and I don’t think I could define a semisimple Lie algebra. Anyway, so what I did was therefore was the easy case: representation theory in transcendental rank. I sort of ended up stumbling into this by accident, so I’ll try to reconstruct the story below, somewhat. I apologize in advance to readers that know algebraic geometry and will probably find this post rather slow-moving (it’s really addressed with a younger version of myself in mind). Readers may wish, however, to review my earlier posts on this topic. […]