In the requests section, Noah asks for help understanding Deligne’s paper “La Categorie des Representations du Groupe Symetrique S_t, lorsque t n’est pas un Entier Naturel.” Peter Arndt refers us to two papers of Knop extending the construction. I’ve been reading them, and I think I understand what’s going on. Moreover, this material is making me think of combinatorial question which seem interesting to me, although others may already have thought about them. I should warn everyone that I had never come across this material until Noah’s question, so there is a risk that I will say something completely ignorant. I also have made no attempt to make my notation match anyone else’s, because I don’t like their notation; if someone wants to compile a dictionary, that would be great.
Nonetheless, here is my attempt at explaining Deligne’s construction in English, with only low level category theory, and with pictures!
Let be a parameter, which for the moment I want to think of as a very large integer. Let be the symmetric group of permutations of letters. Let be the standard -dimensional representation of , over some ground field . Let be the category whose objects are the vectors spaces , , , , … and where is the equivariant homorphisms from to . Tensor products, duals and traces are defined in the obvious way on . (Every object of is self-dual.) By the way, the stands for combinatorial — these are the representations of that are combinatorially obvious.
Let’s try to give a combinatorial description of the category . First of all, let’s find a basis for the vector space . This is the space of invariants in . Now, the dimension of is . It has an obvious basis which I’ll label where each of , , …, , , , …, is an index from .
The group permutes the basis elements , and we get an -invariant for each orbit. For example, when and , the space of -invariants is five dimensional, with basis:
Crucial Difficulty: Well, almost. If is too small, some of these can be zero. For example, vanishes for , and all but vanish when . For now, we’ll think of as very large, and ignore this issue.
In general, we get one basis element of for each equivalence relation on . So the dimension of is given by the -th Bell number. Pictorially, we can represent the basis elements above by figures:
It turns out to be useful to work with a slightly different basis. This basis will also be indexed by equivalence relations on , but we will only impose equalities in the sum, not inequalities. Continuing with the , example, our basis is
Note that the -basis can be expressed by an upper-triangular matrix in terms of the basis. For example,
The entries of the inverse matrix are called the Mobius function of the partition lattice and are given by a certain product of factorials. Pictorially, we will represent the ‘s by the same pictures as the ‘s, but we will use black ink instead of azure.
Now, how do we multiply the ‘s? That is to say, given in and in , how can we expand in the -basis for ? I urge you to do a few examples, and then check them against the following description:
We’ll abbreviate , and . Let and be equivalence relations on and ; let be the equivalence relation they generate on . Let be the number of equivalence classes in which have no representatives in or . Then
Key Equation: .
This is probably easier to explain with a picture:
The factor of comes from the blob on the left, which is an equivalence class containing only elements of .
Now, I can explain what Deligne does when is not an integer. We define to be alatex k$-linear category whose objects are finite sets. is the -vector space with basis the set of equivalence relations on , and composition is by the Key Equation. Note that I write when I am actually thinking of representations of , and for the category which exists for any value of . There is an obvious way to define tensor products and duals, and that gives us, by general nonsense, a way to define a trace.
I don’t really understand that general nonsense, so I’ll tell you the result: let be an equivalence relation on , an equivalence relation on and so forth, with an equivalence relation on . Let be the equivalence relation they generate on , and the number of equivalence classes of . Then the trace of is .
Pictorially, take the pictures of black blobs I’ve been drawing to encode , , …, and wrap them around a cylinder. Let be the total number of blobs you get. Then the trace is .
This is our basic construction; we now must address two technical issues. These are largely independent of each other so, if you don’t understand the next section, feel free to skip to the one after it.
Passing to the Karoubi envelope
We now have defined the category . But we would rather have an analogue of — the category of all representations of . We can already feel the influence of hiding in . For example, consider the ring . (We will return to this example several times.) It is spanned by two elements, , which is the multiplicative identity, and . We have . Set . (Assuming is invertible.) Then is idempotent, meaning that . In most nice categories, having an idempotent in would allow us to split as .
Indeed, when is an integer, has such a splitting in . The -representation has a trivial summand, and is precisely the projection onto this summand. Exercise: Check this!
In order to build our analogue of , we will discuss a construction due to Max Karoubi. In any category, let be an object and an element of satisfying . By definition, an image of is an object with morphisms and such that is the identity and is . It is a pleasant exercise that, if is an image of then, for any other object , we have canonical bijections
Using these equations, a standard argument shows that images are unique up to unique isomorphism. What is more interesting, though, is that there is a way to formally adjoin an image for any idempotent! Given , one formally adds an object to the category, and uses the above equations to define the morphisms to and from . There are a lot of details to check, but they all work out. Given any category, the Karoubi envelope of that category is the result of formally adding images for all idempotents. (You don’t need to iterate the procedure; taking the Karoubi envelope might create some new idempotents, but they will already have images.)
So, Deligne’s analogue of , when is an arbitrary parameter, is the Karoubi envelope of . I’ll denote this by .
The Quotient by the Radical
So far, we have avoided talking about the Crucial Difficulty above. The time has come to face it — for every positive integer value of , the categories and are inequivalent! We can already see the difficulty in the case of . When , we noticed above that we should have . In the -basis, this says or, using the notation we introduced above, .. Similarly, when $t=2$, we should have , which gives a certain nontrivial linear relations between the ‘s in .
If we fix and only consider the subcategories of and induced by the objects , , , …, then, for sufficiently large, these subcategories will coincide. But, no matter how large gets, there will always be some relations between the ‘s in that we don’t see in .
If we think about and instead of , things are worse; we actaully get objects in one category that aren’t in the other. One can (and should!) check that is idempotent in , so it has an image in . In , this image is isomorphic to the zero object, but in it is not.
All of these issues can be fixed by a general construction. I’m not clear on what generality this works in. I believe the details are in “Nilpotence, radicaux et structures monoïdales“, but I haven’t read that paper.
Consider any category with tensor products, duals and (hence) traces. I’m not sure what axioms it should obey, so just think that it should be something like . For any objects and , define to be the set of such that for all . The collection of , as runs through all pairs of objects in , is called the radical of . The radical is a two-sided ideal, meaning that is closed under addition and, for any and any , the product is in . This means that we can form a quotient category , which has the same objects, but where . Any computation with traces will give the same result in and , but some morphisms will be equal in that were unequal in .
Let’s see how differes from . (It doesn’t matter whether we do this computation in or .) The vector space is two dimensional, with basis and . Let’s write for . We have (check!)
When , , which matrix is nonsingular, so . But when , then is in . Thus, in , we have , which is what should happen for actual representations of .
It turns out that, when is not a nonnegative integer, . Moreover, this is a semi-simple abelian category where the isomorphism classes of simple objects are in bijection with partitions (if I understand correctly). When is a nonnegative integer, then is a nontrivial quotient of , and is isomorphic to .
Deligne builds a family of categories, . It is a family in the sense that the objects and morphisms can be viewed as independent of , while the multiplication maps vary polynomially with . When is not a nonnegative integer, the category has no radical. For a nonnegative integer, there is a functor whose kernel is precisely the radical of . Here is the subcategory of the representation category of spanned by the tensor powers of .
By the Karoubi envelope construction, Deligne also builds a category which gives us the ability to split into summands. This can also be thought of as a family of categories in some sense, although I don’t know the details because I am not sure how the Karoubi construction plays with taking limits. (Maybe I’ll post some more about this, if people are interested.) Again, has no radical for not an nonnegative integer. When is a nonnegative integer, there is a functor whose kernel is precisely the radical of .