On uniqueness of tensor products of irreducible categorifications
In this paper, we propose an axiomatic definition for a tensor product categorification. A tensor product categorification is an abelian category with a categorical action of a Kac-Moody algebra in the sense of Rouquier or Khovanov-Lauda whose Grothendieck group is isomorphic to a tensor product of simple modules. However, we require a much stronger structure than a mere isomorphism of representations; most importantly, each such categorical representation must have a standardly stratified structure compatible with the categorification functors, and with combinatorics matching those of the tensor product.
With these stronger conditions, we recover a uniqueness theorem similar in flavor to that of Rouquier for categorifications of simple modules. Furthermore, we already know of an example of such a categorification: the representation category of an algebra previously defined by the second author using generators and relations. Next, we show that tensor product categorifications give a categorical realization of tensor product crystals analogous to that for simple crystals given by cyclotomic quotients of KLR algebras.
Examples of such categories are also readily found in more classical representation theory; for finite and affine type A, tensor product categorifications can be realized as quotients of the representation categories of cyclotomic -Schur algebras.
A subject that has attracted great attention in recent years is that of categorical representations of a Kac-Moody Lie algebra ; this is the study of 2-categories corresponding to universal enveloping algebras of Lie algebras, and in particular, their actions on categories. This theory has deep roots, but the notion of a categorical action of was first introduced by Chuang and Rouquier [CR04], and broadened to other Kac-Moody algebras and developed further by Khovanov and Lauda [Lausl21, KLIII] and Rouquier [Rou2KM].
Obviously, one important question is the relationship between categorical representations and the usual linear representations of . For simple representations, this relationship is quite direct: each simple linear representation of has a universal categorification, by work of Rouquier [Rou2KM]. In essence, simple linear representations and simple categorical representations are in bijection.
However, categorical representations are not necessarily semi-simple, even if the representation on their Grothendieck group is. Examples show that interesting non-irreducible representations should have categorifications which are “more than the sum of their parts”; in this paper, our main example is tensor products of irreducibles, but the same principle applies to the categorifications of Fock space supplied by the categories over Cherednik algebras [ShanCrystal].
In particular, the second author defined a category attached to a list of highest weights in [WebCTP]; in this paper, we will denote this category . This carries a categorical action of and has Grothendieck group isomorphic to the tensor product of modules (or the corresponding representations of if one incorporates the grading). Many properties of this category suggest that it is the “right” categorification for tensor products; in particular, the ribbon structure on the category of modules, and the canonical basis of Lusztig both have appropriate categorical analogues. However, the definition given in [WebCTP] is ad hoc, defined by generators and relations, and lacks a universal property. In this paper, we try to correct this defect, giving an axiomatic characterization of this category, suggested by the notion of a highest weight categorification introduced by the first author [LoHWCI, LoHWCII].
In the crudest sense, the category is the unique abelian category which carries a categorical -action and which is obtained by beginning with the unique categorification of as an irreducible -module, and then adding in new extensions between the projectives in this category in a controlled way. We require that the resulting category is standardly stratified, that is, it has a subcategory of standardly filtered modules which is closed under categorification functors, and whose combinatorics are controlled by those of the tensor product (in particular, the preorder used in the stratification depends on the order of the tensor factors). We formalize this idea with the definition of a tensor product categorification (§3.2). The main body of the paper is dedicated to the proof of the uniqueness result described above:
Theorem A (Thm. LABEL:main2, Thm. LABEL:Thm:cryst).
Any tensor product categorification for the representation is strongly equivariantly equivalent to as a categorical -module.
For any such categorification, there is a canonical isomorphism of crystals between the isomorphism classes of simple objects and the tensor product crystal (as conjectured in [WebCTP, 3.12]).
The proof is by induction; we use the notion of categorical splitting, introduced by the first author in [LoHWCII]. Roughly, inside any categorification of , one finds a categorification of . The inductive hypothesis allows us to identify this subcategory with ; we can then argue that any tensor product -categorification containing an appropriately embedded copy of must be .
This theorem has quite powerful applications in the theory of Lie superalgebras, which will be explored further is a forthcoming joint paper of Brundan and the authors [BLW].
We should note that this theorem is likely not the last word in the question of how one can categorify a tensor product. Of particular import is work announced by Rouquier, which proposes a notion of internal tensor product for the 2-category of categorical -actions. Obviously, we anticipate that our tensor product categorifications are the categories we would arrive at using Rouquier’s internal tensor product, but this remains to be confirmed.
We would like to thank Jon Brundan for his remarks on the previous version of this paper.
I.L. was supported by the NSF under Grant DMS-1161584. B.W. was supported by the NSF under Grant DMS-1151473.
2. Standardly stratified categories
Let be a field of any characteristic. We consider an abelian category such that each block of is equivalent to the category of finite dimensional modules over a finite dimensional -algebra, and such that for every simple , i.e. every irreducible in is absolutely irreducible. We will consider a set with fixed bijection to the isomorphism classes of simple objects in . For any , we let denote the corresponding simple and let be its projective cover.
Now consider a poset with the sets and finite for each . Choose a map with finite fibers; this map induces a natural preorder on . To each , we assign the Serre subcategories (resp., ) of spanned by with (resp., ). Of course, if , then . For , set . For let denote the simple object in corresponding to . Let denote the projective cover of in .
Let denote the quotient functor . We suppose that this functor has an exact left adjoint functor.
We will call this left adjoint the standardization functor and denote it by . We will often omit from the notation. For let (resp., ) denote the object (resp., ). The objects will be called standard and proper standard.
We call the category equipped with a filtration (such that is an exact functor) a standardly stratified category if there is an epimorphism whose kernel admits a filtration by objects with .
If, for each , coincides with the category of vector spaces, then we arrive at the usual definition of a highest weight category.
We remark that, by the definition of , there is an epimorphism and the head of is simple and coincides with . We also remark that the simple constituents of the radical of are of the form with .
It is a standard fact that the condition on a filtration of projectives implies
Let (resp., ) denote the full subcategory of consisting of all objects admitting a filtration whose successive quotients are standard (resp., proper standard) objects. So, in particular, . The following lemma is a direct corollary of (2.1).
Let be the inclusion functor and be its left adjoint functor . Then the functor is exact on (meaning that it maps exact sequences to exact sequences).
For any standardly stratified category , we can consider its associated graded ; we can view as a faithful inclusion , which fails to be full.
For an ideal in (i.e., implies ), we can consider the Serre subcategory spanned by with and the quotient . Both these categories have a natural standardly stratified structure.
The quotient functor has a left adjoint which is exact on . For each , this functor satisfies , where is the standardization functor for . This is an easy corollary of the triangularity property for the projectives. In particular,
2.2. Costandard objects
The category has a “dual” standardly stratified structure, which we describe here. Let denote the injective hull of .
Define the costandard object as the maximal subobject of whose simple constituents are of the form with . Also define the proper costandard object as the maximal subobject of such that the simple subquotients of are of the form with .
Let (resp., ) denote the full subcategories of consisting of all objects admitting a filtration whose successive quotients are costandard (resp., proper costandard) objects.
We have . Moreover, is injective in .
For , we have (resp., ) if and only if (resp., ) for all .
The right adjoint functor of the projection is exact and
Here we write for the injective envelope of in . We remark that here it is essential that we require the standardization functor to be exact, see [CPS96] for a counter-example.
Part (1) is a standard calculation using the filtrations on projective and injective modules. Thus we turn to part (2); furthermore, since the proofs are parallel, we only check that if and only if . By induction, we may assume that and that for objects in , our claims are proved.
Let be the largest subobject in belonging to . Then for any with we have . Therefore, by the inductive assumption, is -filtered and hence for all . It follows that . So it is enough to consider the case when , i.e., the socle of is a sum of simples of the form for . It follows that embeds to the sum of several ’s with such that the socles of and of coincide. But for such we have . If we have for all , then we have a surjection . But since is the sum of costandard objects, all homomorphisms from to factor through the socle of and hence through . So for all and therefore . Thus, has a costandard filtration.
Part (3) follows immediately; consider an exact sequence in . We can assume by induction that both are -filtered. The cokernel of satisfies for all and so is -filtered. So we have an embedding of -filtered objects that becomes an isomorphism after projecting to . This forces the costandard quotients of to be the same and the embedding to be an isomorphism (recall that all blocks of are finite). ∎
We also point out the following form of the BGG reciprocity.
The multiplicity of in equals to the multiplicity . Similarly, the multiplicity equals .
Combining these results, we see that:
The category is standardly stratified with respect to the map and the standardization functor .
For highest weight categories, one can define tilting objects and has the Ringel duality. This can be generalized to standardly stratified categories. Namely, we say that an object in a standardly stratified category is tilting if it is both -filtered and -filtered (there is also a dual notion of a co-tilting object that has to be - and -filtered). Then, similarly to the highest weight case, one can show that every standard object has a unique tilting hull ; the ’s are pairwise non-isomorphic and exhaust all indecomposable tiltings. Set . By the Ringel dual of we mean the category of finite dimensional right -modules. This category admits a natural standardly stratified structure with standardization functor . One can check the axioms similarly, for example, to [RouqSchur, 4.1.5]. We have a natural equivalence .
3. Tensor product categorifications
3.1. Categorical -actions
Let be a Kac-Moody algebra with its set of simple roots. There are a variety of notions of categorical -actions which have appeared in the literature, in the work of Rouquier [Rou2KM], Khovanov and Lauda [KLIII], Cautis and Lauda [CaLa] and others. Of course, as with all definitions where there is some flexibility, one endeavors to use the weakest version possible when proving facts about objects satisfying said definition and the strongest when showing that an object does satisfy it (though one is often forced to do the opposite).
All of these definitions employ the KLR algebra or quiver Hecke algebra , a sum of finitely generated algebras attached to every element in the positive cone of the root lattice of ; we let for an integer denote the sum of the ’s for a sum of simple roots. We should note that our definition of KLR algebra follows that of Rouquier [Rou2KM, §3.2], and thus involves a choice of polynomials for each pair of elements in the Dynkin diagram with degree in bounded above by the entry of the negated Cartan matrix, and similarly the degree in bounded above by . We say this choice is homogeneous if the polynomial is homogeneous when the ratio between the degrees of the variables and equals the ratio between the lengths of the simply roots and . We’ll follow the conventions of [WebCTP, §1] throughout.
The finitely generated modules over the KLR algebra form a monoidal category under induction functors; this monoidal category on its own is a categorification, in a certain sense, of the enveloping algebra of the Borel .
For our purposes, a categorical -action on an additive category is an action of the strict 2-category Rouquier denotes ; that is, it consists of
a module category structure over the representations of the KLR algebra generated by functors ; that is, a functor such that carries an action of . In particular, each of the functors carries a natural transformation , usually denoted as a dot in literature such as [KLIII, WebCTP, KI-HRT], and
right adjoints to these functors, such that
the map Rouquier denotes in [Rou2KM, §4.1.3] is an isomorphism.
In particular, each pair of functors should be thought of a categorical action in the sense of Chuang and Rouquier (in fact, they only consider categories which are abelian, artinian and noetherian, so small adjustments in the definition would be necessary). All other notions of categorification mentioned above are adding additional structure to this schema.
Since our main theorem will be a classification/uniqueness theorem, we need to have a notion of equivalence between categorical actions.
A strongly equivariant functor between two categories with categorical -actions is
a functor together with
isomorphisms of functors which commute with the -actions on .
If we think of a categorical -action as a representation of the 2-category in the strict 2-category of -linear categories, this is the usual notion of natural transformation between representations of a 2-category.
We call such a functor a strongly equivariant equivalence if is an equivalence.
Our starting point is the categorification of , an irreducible representation of with highest weight . As mentioned in the introduction, there are several uniqueness theorems for such categorifications, based on work of Rouquier [Rou2KM, §5.1]; since there are different contexts in which such representations appear, we record here the version that we require. We use to denote the cyclotomic KLR algebra for and the highest weight (for example, as discussed in [WebCTP, §1]), and let to denote the finite dimensional subalgebra spanned by diagrams with strands, that is, the image of .
Assume is an artinian abelian -linear category.
Assume has a -action by exact functors such that:
the Grothendieck group isomorphic to ;
the subcategory has a fixed equivalence to , sending to an object ;
the transformation acts nilpotently on .
Then is strongly equivariantly equivalent to the category of finite-dimensional modules over . The adjoint equivalences are given by
The definition of a categorical action shows that the projective objects carry actions of the KLR algebra . Note that we must have
In fact, the categorification conditions imply that the monomials span the algebra ; this is a special case of [CaLa, 3.12]. Since is nilpotent, we must have
This implies that the cyclotomic ideal for acts trivially on .
Thus, if we consider the universal strongly equivariant functor from Rouquier’s category to , it factors through the category of projective -modules, so the induced functor is strongly equivariant. Furthermore, since this functor is fully faithful on , this establishes that projective -modules are the desired base change; in Rouquier’s notation . Thus, this functor is fully faithful by [Rou2KM, 5.4].
So, in order to show that we have an equivalence, we need only show that every indecomposable projective is a summand of . Since summands of obtained using different monomials in the applied to span the Grothendieck group , there can be no others and we are done. ∎
We’ll note, this implies that in , every irreducible is absolutely irreducible, since this holds for the KLR algebra.
Let be irreducible integrable -modules with highest weights and be an infinite field; we let . We are going to define the notion of a categorification of the ordered tensor product .
As before, let be an abelian artinian -linear category, with each block equivalent to the representation category of a finite dimensional -algebra. The data of a tensor product categorification on consists of two parts:
a categorical -action on in the sense of Rouquier where the functors and are exact, and the natural transformation acts locally nilpotently on and,
the structure of a standardly stratified category on with poset .
These two pieces of data have to satisfy some compatibility conditions to be explained below.
The poset is the set of -tuples , where is a weight of . The poset structure is given by “inverse dominance order”:
As usual, we write if is a linear combination of positive roots with non-negative coefficients. We should note that the importance of order becomes immediately apparent in this definition.
The associated graded category carries a categorical -action with as -modules such that the weight subcategory of is precisely the subquotient . We use and to denote categorification functors for the th copy of . We assume that ; we let denote the unique indecomposable object in this subcategory. By Proposition 3.2, we thus have that , the representations of the cyclotomic KLR algebra of for the highest weight .
Finally, we must have a compatibility between the categorical -action on and the categorical -action on : for each , the object admits a filtration with successive quotients being . It is easy to see that such a filtration is determined uniquely, we call it a standard filtration on .
Similarly, we require that comes equipped with a filtration whose successive quotients are .
Since every irreducible is absolutely irreducible in as we noted above, this means that the same property will hold in any tensor product categorification.
Of course, we could try to make this definition more general by not requiring to be nilpotent; however, this would not really gain us any additional generality. If acted on the functor with more than one eigenvalue, it could then be split into generalized eigenspaces for , and these functors would give a pair of categorical actions of . Thus, we may as well assume that has only one eigenvalue . Note that we can change this eigenvalue using the substitution , at the cost of changing the relations of the KLR algebra. We must change the polynomials by the same substitution.
Another point where the reader might wish to generalize this is to replace the condition that with the condition that (say) is the representation category of a local Artinian -algebra . Our results should extend to this case, but at a considerable cost; in particular, the categorification obtained is no longer unique. Rather, the possible choices of will have moduli, given by considering the minimal polynomial of for its induced action on ; the coefficients of this polynomial can be arbitrary elements of the radical of , and one expects that there is a unique TPC with this choice of . Aside from the intrinsic nuisance of working relative to , there are two relatively minor, but non-trivial, technical obstacles here:
there are competing definitions of categorical -action, and it’s not clear that they give the same result. The classification mentioned above in terms of minimal polynomials is known for the Cautis-Lauda 2-category from [CaLa] by [WebCTP, 1.12]; Rouquier has announced the same result for his 2-category, but the proof has yet to appear.
it is not actually proven that a TPC will exist in this relative case since the corresponding algebras are not considered in [WebCTP], though most results could be ported over by a careful use of Nakayama’s Lemma.
As in any standardly stratified category, we have an isomorphism of Grothendieck groups via the map sending . By assumption, we obtain an isomorphism . It follows immediately from (TPC3) that:
For any tensor product categorification, this map is an isomorphism of -modules.
We also note that any tensor product categorification is integrable, so by [Rou2KM, 5.16], the functors and are necessarily biadjoint.
In fact, we could give an axiomatic description of a tensor product of arbitrary -categorifications . Let us elaborate on this in the case when . We say that a -categorification equipped with a standardly stratified structure is the tensor product if
the poset of is the set of pairs , where is a weight for , with the order given by if and .
there is an identification of with
for each the object admits a short exact sequence
and similarly for the functors .
Unfortunately, in general, we can prove neither existence nor uniqueness of such tensor products; we expect that it will arise from Rouquier’s proposed internal tensor product.
Before getting too deep into the structure of these categories, we should give a set of labels for simples (or indecomposable projectives) in .
As usual with a standardly stratified category, the simples (or indecomposable projectives) in are in canonical bijection with the simples (or indecomposable projectives) in . Let be the crystal of the irreducible representation . By [LV, §5.1], we have a canonical bijection between the product and the set of simples (or indecomposable projectives) in . Recall that the set of simple objects in an arbitrary -categorification has a -crystal structure: if is a simple, then for we take the head (equivalently, the socle) of the object if the latter is nonzero and else. The crystal operator is defined similarly using ; as usual, we use the notation when considering these operators for .
First consider the case of the categorification of a simple module with highest weight , which, as before, we denote . One straightforward description of the projective for a crystal element uses the string parameterization of vertices of . Consider for some , and let be the associated projective. Choose an infinite sequence of nodes in the Dynkin diagram of containing each node infinitely many times. The string parameterization of is the unique infinite sequence of integers with almost all entries 0 such that
We can order crystal elements by comparing string parametrizations lexicographically.
Proposition 3.7 (Khovanov-Lauda [Kli, 3.20]).
The projective is the unique summand of which doesn’t appear in for a word larger than in lexicographic order.
For , we want to proceed a little differently; instead of applying this construction directly (which will work perfectly well), we compute the string parameterization of each component of . Thus, we obtain different words , etc. We can easily modify the proposition above to:
The projective is the unique summand of which doesn’t appear in a corresponding monomial where any is replaced by a larger word in lexicographic order.
3.4. Tensor product categorification on
Suppose that is a tensor product categorification of in the sense of the definition above. In this subsection we are going to prove that is also a tensor product categorification of .
The category is a tensor product categorification of
Conditions (TPC1) and (TPC2) are tautologically equivalent for and . Thus, we need only establish (TPC3).
We need to prove that has a filtration whose successive quotients are and the analogous claim for . Here means the simple root for the th copy of .
First of all, preserve . This follows from the observation that preserve , combined with Lemma 2.4 and the biadjointness of and .
Pick , where . We see that
where all equalities are natural isomorphisms of -modules.
Now let us show that the claim in the beginning of the proof holds when is projective (=injective) in . Recall that . In particular, if, in (3.1), for we take the simple in labeled by , we see that the multiplicity of in and coincide. This implies the existence of a required filtration on .
Proceed to the case of a general . In this case, Lemma 2.4 just implies that . So, by (2.1), the object has a filtration with successive quotients , , for . Recall that we write for the inclusion functor , for the projection functor , and for the left adjoint functors. Choose and set . Consider the functor . It follows from Lemma 2.2 that the functor is exact on . It maps to the subquotient of the form , where . So we just need to prove that the functor is isomorphic to if for some and is zero else. The vanishing result follows from the form of , for projective, obtained above. The isomorphism of functors follows from (3.1). ∎
One can also equip the Ringel dual with a categorical -action turning into a tensor product categorification of , compare with [LoHWCII, 7.1] and [Lotowards, 9.2]. Namely, using the identification one defines the categorification functors on and then extends them to the whole category obtaining a categorical -action. Then it is not difficult to see that together with the standardly stratified structure on , this action becomes a tensor product categorification of .
3.5. Relation with previous constructions
Concrete examples of categorical -actions whose Grothendieck groups are tensor products have arisen in work in representation theory and topology.
One obvious construction to compare the definition above with are the algebras defined by the second author in [WebCTP, §2]. We refer the reader to that paper for the details of the definition. What is important for us is an inductive description of the representation categories of these algebras. Given the sequence of weights , we define weights .
Attached to each , we have an associated cyclotomic quotient of the KLR algebra
equipped with projections , and induced inflation functors .
Now, consider the category defined as the category of representations of its category of projectives (via the Yoneda embedding):
we let just be the category of finite dimensional representations of .
The category of projectives is the additive category of generated by summands of categorification functors applied to the image .
Thus, these are the minimal subcategories closed under the categorical -action which contains the images of all inflation functors.
We have an equivalence .
This follows from [WebCTP, 3.24]. In that paper we define a fully faithful functor whose essential image is exactly the additive category generated by summands of for certain elements associated to sequences and functions . If , then where . Thus, the image of is generated by categorification functors applied to the modules with . Since these modules are exactly obtained by applying the inflation functor to the images of , we are done. ∎
The most important fact for us is that:
The category is a tensor product categorification for .
This category is a categorical -module by [WebCTP, 2.11] and standardly stratified by [WebCTP, 3.18]. (TPC1) is also part of the statement of [WebCTP, 3.18], (TPC2) follows from [WebCTP, 3.21] and (TPC3) follows from [WebCTP, 3.7-8]. ∎
Tensor product categorifications for also arise in more classical representation theory. Here, we give two examples.
Consider the Lie algebra , its parabolic subalgebra with blocks (from top to bottom) of sizes and also fix a positive integer . Let be the corresponding parabolic category . The integral blocks of this category form a highest weight category whose standard objects are parabolic Verma modules
of highest weight .
Let be the the sum of blocks of spanned by with .
This category is highest weight in the sense of [LoHWCII] (more precisely, all categorical -actions corresponding to the simple roots are highest weight) and so is a tensor product categorification of . It was shown in [WebCTP, 4.2] that the category is strongly equivariantly equivalent to , where is the th fundamental weight. The main theorem of this paper also provides a new proof of this equivalence.
Often we can also realize tensor product categorifications as subquotients of interesting categories. Let us give an example when the field has characteristic and the algebra acting is . Consider the category , where is the category of polynomial representations of of degree . This is a highest weight category, whose labeling poset is that of partitions (with respect to the -dominance ordering).
A categorical -action on this category was first introduced in [HY], and this action is highest weight in the sense of [LoHWCII]. Fix a residue and consider the subalgebra corresponding to the other residues.
We introduce an equivalence relation on the set of Young diagrams: if the boxes in and with residue are the same. Attached to each such equivalence class is a list of coordinates given by the rightmost box in each diagonal of the partition diagram with content congruent to listed left to right; we must also include the first empty diagonals encountered on the left and right, that is, we have and . We let , and note that .
Each equivalence class is an interval in the highest weight poset of , so for an equivalence class , we can consider the subquotient category corresponding to . This is a highest weight category with a well-defined highest weight categorical action of and a tensor product categorification of the product .
For example, if and
, we have the sequence of boxes
and the tensor product categorified is .