The Khovanov-Lauda 2-category and categorifications of a level two quantum sl(n) representation

We construct 2-functors from a 2-category categorifying quantum sl(n) to 2-categories categorifying the irreducible representation of highest weight $ 2 \omega_k. $


introduction
Khovanov and Lauda introduced a 2-category whose Grothendieck group is U q (sl n ) [11]. This work generalizes earlier work by Lauda for the U q (sl 2 ) case [13]. Rouquier has independently produced a 2category with similar generators and relations, [15]. There have been several examples of categorifications of representations of U q (sl n ) arising in various contexts. Khovanov and Lauda conjectured that their 2-category acts on various known categorifications via a 2-functor. For example, in their work they construct such a 2-functor to a category of graded modules over the cohomology of partial flag varieties. This 2-category categorifies the irreducible representation of U q (sl n ) of highest weight nω 1 where ω 1 is the first fundamental weight.
In this note we construct this action for the categorification constructed by Huerfano and Khovanov in [8].
They categorify the irreducible representation V 2ω k of highest weight 2ω k , by a modification of a diagram algebra introduced in [9]. The objects of 2-category HK k,n are categories C λ which are module categories over the modified Khovanov algebra. We explicitly construct natural transformations between the functors in [8] and show that they satisfy the relations in the Khovanov-Lauda 2-category giving the theorem: Theorem. There exists a 2-functor Ω k,n : KL → HK k,n .
The Huerfano-Khovanov categorification is based on categories used for the categorification of U q (sl 2 )tangle invariants. This hints that a categorification of V 2ω k may also be obtained on maximal parabolic subcategories of certain blocks of category O(gl 2k ). More specifically, we construct a 2-category P k,n whose objects are full subcategories Z P (k,k) µ (gl 2k ) of graded category Z O (k,k) µ (gl 2k ) whose set of objects are those modules which have projective presentations by projective-injective objects. The 1-morphisms of P k,n are certain projective functors. We explicitly construct the 2-morphisms as natural transformations between the projective functors by the Soergel functor V. We then prove: Theorem. There is a 2-functor Π k,n : KL → P k,n .
It should be possible to categorify V N ω k for N ≥ 1 using categories which appear in various knot homologies. For N ≥ 2, the module categories C λ in the Huerfano-Khovanov construction should be replaced by suitable categories of matrix factorization based on Khovanov-Rozansky link homology. The categories of matrix factorizations must be generalized from those used in [12]. Khovanov  categories of matrix factorizations should be taken over tensor products of polynomial rings invariant under the symmetric group. These categories were studied in depth by Yonezawa and Wu [21,20]. In fact, the isomorphisms of functors categorifying the U q (sl n ) relations were defined implicitly in [20]. To check that there is a a 2-representation of the Khovanov-Lauda 2-category, these isomorphisms would need to be made more explicit. The category O approach should be modified as well. Now the objects of the 2-category should be subcategories of parabolic subcategories corresponding to the composition N k = k + · · · + k of blocks of O λ (gl(N k)), and the stabilizer of the dominant integral weight µ is taken to be S λ1 × · · · × S λn where each λ i ∈ {0, 1, . . . , N }, cf. Section 5 below. Note that a categorification of V λ for arbitrary dominant integral λ, hence in particular of V N ω k , is constructed in [4] using cyclotomic quotients of Khovanov-Lauda-Rouquier algebras.
While this paper was in preparation, two very relevant papers appeared. In [6], J. Brundan and C. Stroppel also defined the appropriate natural transformations and checked relations between them to establish a version of the first theorem above, but for Rouquier's 2-category from [15] rather than the Khovanov-Lauda 2-category. One of the advantages of their result is that they are able to work over an arbitrary field, while we work over a field of characteristic 2. It is not immediately clear to us how to use their sign conventions to get an action of the full Khovanov-Lauda 2-category in characteristic zero, because they seem to lead to inconsistencies between propositions 3, 4, 6, and 12. Additionally, Brundan and Stroppel categorify V 2ω k using graded category O. More precisely, they first categorify the classical limit of V 2ω k at q = 1 using a certain parabolic category O, without mentioning gradings. Then they establish an equivalence between this category and the (ungraded) diagrammatic category. Finally, they observe that both categories are Koszul (by [1] and [5], respectively) so, exploiting unicity of Koszul gradings, their categorification at q = 1 can be lifted to a categorification of the module V 2ω k itself in terms of graded category O. Our construction on the graded category O side is more explicit, relying heavily on the Soergel functor, the Koszul grading that O inherits from geometry, and explicit calculations on the cohomology of flag varieties made in [11]. In the other relevant paper, M. Mackaay [14] constructs an action of the Khovanov-Lauda 2-category on a category of foams which is the basis of an sl 3 -knot homology.
Acknowledgements: The authors would like to thank Mikhail Khovanov and Aaron Lauda for helpful conversations.
2. The quantum group U q (sl n ) 2.1. Root Data. Let sl n = sl n (C) denote the Lie algebra of traceless n×n-matrices with standard triangular decomposition sl n = n − ⊕ h ⊕ n + . Let ∆ ⊂ h * be the root system of type A n−1 with simple system Π = {α i |i = 1, . . . , n − 1}. Let (·, ·) denote the symmetric bilinear form on h * satisfying where A = (a ij ) 1≤i,j<n is the Cartan matrix of type A n−1 : Let ∆ + be the set of simple roots relative to Π. Let ω 1 , . . . , ω n−1 ∈ h * be the elements satisfying (ω i , α j ) = δ ij , and let Zω i , and P + = the definition of a ij to all i, j ∈ I accordingly. Finally, for i ∈ I, let sgn(i) = i/|i| be the sign of i.
The quantum group U q (sl n ) is the associative algebra over Q(q) with generators E i , K i , for i ∈ I satisfying the following conditions: We fix a comultiplication ∆ : U q (sl n ) → U q (sl n ) ⊗ U q (sl n ) given as follows for all i ∈ I + : Via ∆, a tensor product of U q (sl n )-modules becomes a U q (sl n )-module.
In this paper we are interested in the irreducible U q (sl n )-modules, V 2ω k with highest weight 2ω k . Therefore, we will identify the weight lattice P ∼ = Z n−1 ⊂ Z n as follows: Assume λ = i a i ω i . For each 1 ≤ i < n set Let P (2ω k ) denote the set of weights of V 2ω k . It is well known that under this identification each λ ∈ P (2ω k ) satisfies λ i ∈ {0, 1, 2} for all 1 ≤ i ≤ n and λ 1 + · · · + λ n = 2k.

The Khovanov-Lauda 2 category
Let k be a field. The k-linear 2-category KL defined here was originally constructed in [11]. The original construction is defined conveniently in terms of diagrams. We do not present the generators and relations in terms of diagrams here because it would conflict with the diagrams used in the construction of the 2-representation in the next section.
3.1. The objects. The set of objects for this 2-category is the weight lattice, P .
3.2. The 1-morphisms. For each λ ∈ P , let I λ ∈ End KL (λ) be the identity morphism and, for λ, λ ′ ∈ P , . . , i r ) ∈ I ∞ , and s refers to a grading shift. Observe that 3.3. The 2-morphisms. The 2-morphisms are generated by to be the identity transformation.
For convenience of notation, we define the following 2-morphisms.
For each i ∈ I, define the bubble Also, define half bubbles We now define the relations satisfied by these basic 2 morphisms. In what follows, we omit the argument λ when the relation is independent of it.
(c) For i, j ∈ I + , i = j, and Tr(1) = 0. There is also a unit map ι : C → A given by ι(1) = 1. Also, let κ : A → A be given by κ(1) = 0, κ(x) = 1. This algebra gives rise to a two dimensional TQFT F, which is a functor from the category of oriented 1 + 1 cobordisms to the category of abelian groups. The functor F sends a disjoint union of m copies of the circle S 1 , to A ⊗m . For a cobordism C 1 , from two circles to one circle, F(C 1 ) = m. For a cobordism C 2 from one circle to two circles F(C 2 ) = ∆. For a cobordism C 3 , from the empty manifold to S 1 , For a cobordism C 4 from the empty manifold to S 1 , F(C 4 ) = Tr.
For any non-negative integer r, consider 2r marked points on a line. Let CM r be the set of non-intersecting curves up to isotopy whose boundary is the set of the 2r marked points such that all of the curves lie on one side of the line. Then there are (2r)! r!r!(r+1) elements in this set. The set of crossingless matches for r = 2 is given in figure 1. Let a, b ∈ CM r . Then (Rb)a is a collection of circles obtained by concatenating a ∈ CM r with the reflection Rb of b ∈ CM r in the line. Then applying the two dimensional TQFT F, one associates the graded vector space b H r a to this collection of circles. Taking direct sums over all crossingless matches gives a graded vector space This graded vector space obtains the structure of an associative algebra via F, cf. [9].
Let T be a tangle from 2r points to 2s points. Let a be a crossingless match for 2s points and b a crossingless match for 2s points. Then let a T b be the concatenation Ra To any tangle diagram T from 2r points to 2s points, there is a (H s , H r )-bimodule To any cobordism C between tangle T 1 and T 2 , there is a bimodule map F(C) : is the Euler characteristic of C cf. proposition 5 of [9].
Consider the tangles I and U i in figure 4. Then there are saddle cobordisms S i : U i → I and S i : (1) There exists an (H n−1 , H n )-bimodule homomorphism µ i : Proof. There is a degree zero isomorphism of bimodules Then by [9] there is a bimodule map of degree one Then µ i is the composition of these maps.
The construction of µ i is similar.
Lemma 2. Let a ∈ CM n and b ∈ CM n−1 be two crossingless matches. Let T i be the tangle on the right side of the Figure 5. Let U i be the tangle in Figure 4. Consider the homomorphism induced by the cobordism S i , where α ∈ A corresponds to the circle passing through the point i on the top line and β ∈ A ⊗p corresponds to the remaining circles. Then α⊗β → ∆(α)⊗β.
Proof. The map is induced by the cobordism S i . On the set of circles, this cobordism is a union of identity cobordisms and a cobordism C 2 . The result now follows upon applying F. Lemma 3. Let I be the identity tangle from 2r points to 2r points, T i a tangle from 2(r + 1) points to 2r points and T i a tangle from 2r points to 2(r + 1) points. Let a and b be cup diagrams for 2r points. Consider the map where the first and last maps are isomorphisms and the middle map is µ i ⊗ 1. Let β ∈ A correspond to the circle passing through point i of a I b , γ ∈ A ⊗r correspond to the remaining circles and α ∈ A. Then the map Proof. The map is induced by a cobordism S i+1 . On the set of circles, this cobordism is union of identity cobordisms and a cobordism C 1 . The result now follows upon applying F.

The Huerfano
Label n collinear points by the integers λ i . Those points labeled by 0 or 2 will never be the boundaries of arcs but will rather just serve as place holders. Then define the algebra . Let e λ be the identity element.
Let i ∈ I + . We define five special tangles D λ,i , D λ,i , T λ,i , T λ,i , I λ in figures 6, 7, 8. If a point is labeled by zero or two, it will not be part of the boundary of any curve. Away from points i, i + 1 the tangle is the identity. The cobordisms S λ,i : T λ+αi,i • T λ,i → I λ and S λ,i,j : T λ+αi,j • T λ,i → D λ+αj ,i • D λ,j are saddle cobordisms for j = i ± 1. Similarly, the cobordisms S λ,i , S λ,i,j are saddle cobordism in the opposite direction. For example, the cobordism S λ,i,i+1 is given in figure 9.
Let C λ be the category of finitely generated, graded H λ -modules, and let I λ : C λ → C λ be the identity  Figure 11. T λ,−i and T λ,−i Let i ∈ I. Let I λ : C λ → C λ denote the identity functor which is tensoring with the (H λ , H λ )− bimodule H λ . Let E i I λ : C λ → C λ+αi be the functor of tensoring with a bimodule defined as follows: Evidently, E i I λ = I λ+αi E i I λ for all i ∈ I, and I λ = F(I λ ).
For i ∈ I, let K i I λ : C λ → C λ be the grading shift functor Propositions 2 and 3 of [8] are that these functors satisfy quantum sl n relations: Now we define the Huerfano-Khovanov 2-category HK k,n over the field k, chark = 2.
4.3. The objects. The objects of HK k,n are the categories C λ , λ ∈ P (V 2ω k ).
Let i ∈ I, and let 1 i,λ : E i I λ → E i I λ , and 1 λ : I λ → I λ be the identity maps.
For i ∈ I we define maps y i;λ : E i I λ → E i I λ of degree 2. Let T be the tangle diagram for the functor E i I λ . It depends on the pair (λ i , λ i+1 ). Let a and b be crossingless matches such that (Rb)T a is a disjoint union of circles. Thus F((Rb)T a) = (A) ⊗p for some natural number p. Define     We define a map ∪ i;λ : There are four non-trivial cases for (λ i , λ i+1 ) to consider.
(a) (λ i , λ i+1 ) = (1, 2). The identity functor is induced from the identity tangle I λ . The functor E −i E i is isomorphic to tensoring with the bimodule F(D λ+αi,i • D λ,i ) which is equal to F(I λ ).
Thus in this case ∪ i;λ is given by the identity map. 1). Then the functor E −i E i is isomorphic to tensoring with the bimodule (c) (λ i , λ i+1 ) = (0, 2). Then the functor E −i E i is isomorphic to tensoring with the bimodule Then the bimodule map is given by 1 λ ⊗ ι.
(d) (λ i , λ i+1 ) = (0, 1). The functor E −i E i is isomorphic to tensoring with the bimodule F(D λ+αi,i • D λ,i ). As in case 1, this tangle is isotopic to the identity so the map between the functors is the identity map.
We define a map ∩ i;λ : There are four non-trivial cases for (λ i , λ i+1 ) to consider.
(a) (λ i , λ i+1 ) = (1, 2). The functor E −i E i is isomorphic to tensoring with the bimodule F(D λ+αi,i • D λ,i ) which is equal to F(I λ ). Thus in this case ∩ i;λ is given by the identity map.
As in case 1, this tangle is isotopic to the identity so the map between the functors is the identity map.
We define a map ψ i,j;λ : There are four cases for i and j to consider and then subcases for λ.
(a) i = j. In this case, the functors are non-trivial only if λ i = 0 and λ i+1 = 2. The bimodule for E i E i is isomorphic to tensoring with the bimodule F(T λ+αi,i • T λ,i ) = F(I λ ) ⊗ A. Then (b) |i − j| > 1. In this case, the functors E i E j and E j E i are isomorphic via an isomorphism induced from a cobordism isotopic to the identity so set ψ i,j to the identity map.
There are four non-trivial subcases to consider.
In this case we define the bimodule map to be F(S λ,i,i+1 ).
For convenience of notation, we define the following 2-morphisms.
For each i ∈ I, define the bubble and define fake bubbles inductively by the formula  and, • −1 i;λ = 1 whenever (α i , λ) = 0. Also, define half bubbles Finally, for i, j ∈ I ± , define 4.6. The 2-morphism relations. Again, we will often omit the argument λ when it is clear from context.
Proof. The second equality is similar to the first equality. The case i ∈ I − is similar to the case i ∈ I + so we just compute the map (∩ i 1 i ) • (1 i ∪ i ) on the bimodule for the functor E i for i ∈ I + . There are four cases to consider.
Suppose (λ i , λ i+1 ) = (1, 2). Then the tangle diagrams for the functors E i and E i E −i E i are D λ,i and D λ,i • D λ+αi • D λ,i and can be found in Figure 12.
The cobordism between the tangles is isotopic to the identity map so in this case the composition is equal to the identity map.
The case (λ i , λ i+1 ) = (0, 1) is similar to the (1, 2) case. Now let (λ i , λ i+1 ) = (0, 2). Then the tangle diagrams for the functors E i and E i E −i E i can be found in Let B be the bimodule for the functor E i . Then the bimodule for Finally consider the case (λ i , λ i+1 ) = (1, 1). The tangle diagrams for the functors E i and E i E −i E i can be found in Figure 14.
Let B be the bimodule giving rise to the functor E i and A ⊗ B be the bimodule giving rise to the functor where α is in the tensor factor corresponding to the circle passing through point i on the bottom row of the left side of Figure 14 and β belongs to the remaining tensor factors.
The cobordism between the two tangle diagrams is a saddle which on the level of bimodule maps, sends Proof. We prove only the first equality as the second is similar. There are four cases to consider for which the functor E i is non-zero.
Suppose (λ i , λ i+1 ) = (1, 2). Then the tangle diagrams for the functors E i and E i E −i E i can be found in On the other hand, y i (α ⊗ β) = xα ⊗ β.
Suppose (λ i , λ i+1 ) = (0, 2). Then the bimodule for the functor E i is B = F(T λ,i ) and the tangle diagram where α is an element of the tensor factor corresponding to the circle passing through the point i in the top row of the tangle T λ,i and β is an element in the remaining tensor factors. Then the composition of maps send α⊗ β → 1 ⊗ α⊗ β → x⊗ α⊗ β → xα⊗ β.
This is equal to y i (α ⊗ β).
Suppose (λ i , λ i+1 ) = (1, 1). Then the tangle diagrams for the functors E i and E i E −i E i can be found in Figure 14.
Let B be the bimodule for the functor E −i and A ⊗ B be the bimodule for α is an element in the tensor factor corresponding to the circle passing through point i on the bottom row of Figure 14 and β is an element in the remaining tensor factors. First let α = 1. Then where the last map is Tr ⊗ 1. If α = x, then Proposition 5. Suppose i ∈ I and (−α i , λ) > r + 1, then •r i;λ = 0.
Proof. In order for r ≥ 0, it must be the case that (−α i , λ) ≥ 2. Thus the only possibility is (λ i , λ i+1 ) = (0, 2) and r = 0. Then the bimodule for E −i E i is A ⊗ F(I λ ). Thus the map 1 → E −i E i is given by the unit map.
The map E −i E i → 1 is given by the trace map. Thus the composition of the maps in the proposition sends Proof. The only cases to consider are (λ i , λ i+1 ) = (0, 2), (1, 2), (0, 1).
For the case (1, 0), the first term on the right hand side is zero since that map passes through the functor The summation on the right hand side reduces to This composition of maps is the identity.
For the case (2, 0), the first term on the right hand side is zero as in the previous two cases. The summation on the right hand side consists of three terms which simplifies by (1) to: Under this composition of maps, 1 ⊗ b maps to zero since the first map is given by a trace map on the first component. The element x ⊗ b gets mapped to x ⊗ b as follows: where the first map is the trace map, the second map is the unit map and the third map is multiplication by x. Similarly, is zero because the middle term is zero. Thus the right hand side is the identity as well.
(1) If (α i , λ) ≤ 0, then Proof. We prove (1), the proof of (2) being similar. Since the map on both sides pass through the functor given by tangles in Figure 14.
Let B be the bimodule for the functor E i so A ⊗ B is the bimodule for the functor where α is an element in the tensor factor corresponding to a circle passing through point i in the bottom row of the left side of figure 14 and β is an element in the other tensor factors. Consider first α = 1. The left hand side maps an element α ⊗ β as follows: where the first map is ∆ ⊗ 1, the second map is κ ⊗ 1 ⊗ 1 and the third map is m ⊗ 1. If α = x, the left hand maps α ⊗ β as follows: The right hand side is −1 by convention.

Then the bimodule for
Then Proof. Both sides are natural transformations of the functor E i E i E i . However, by definition this composition is zero.
Proof. The only case to check is (λ i , λ i+1 ) = (0, 2) since otherwise E i E i = 0. Let B = F(I λ ). Then the bimodule for E i E i is isomorphic to A ⊗ B. Then This gives the first equality since our field is F 2 .
For the second equality, ( Proof. Let i, j ∈ I − . We prove only the first equality. If |i − j| > 1, the proposition is easy because then ψ ±i,±j are identity morphisms. Therefore, we take i = j + 1, the case i = j − 1 being similar. The natural transformation on the right side of the proposition is a composition of natural transformations: There are four nontrivial cases for λ. We prove the case (λ j , λ j+1 , λ j+2 ) = (2, 1, 1). The proofs of the remaining cases (2, 1, 0), (1, 1, 0), and (1, 1, 1) are similar.  Figure 15. The first and second maps are the identity maps. The third map is comultiplication.
The fourth map is the counit map and the last map is ψ j,j+1 . Computing this composition on elements as in previous propositions easily gives that it is equal to ψ j,j+1 . R(ν) relations.
Proposition 13. For i, j ∈ I ± , i = j, Proof. Note that for |i − j| > 1, the left hand side is easily seen to be the identity so let j = i + 1. The case • Suppose that the circle passing through point i + 1 on the bottom row of a (T λ+αi+1,i ) • T λ,i+1 ) b is the same as the circle passing through point i of the top row. Then a B b = A ⊗ R and a B ′ b = A ⊗ A ⊗ R where R is a tensor product of A corresponding to the remaining circles. Then the map on the left side of the proposition is (m ⊗ 1) • (∆ ⊗ 1). Thus it maps an element 1 ⊗ r to 2x ⊗ r. On the other hand, y i (1 ⊗ r) = +x ⊗ r. Also, y i+1 (1 ⊗ r) = x ⊗ r. Thus both sides are the same.
• Suppose that the circle passing through point i + 1 on the bottom is different from the circle passing through point i on the top. Then a B b = A ⊗ A ⊗ R and a B ′ b = A ⊗ R. Then the map on the left side of the proposition is (∆ ⊗ 1 λ ) • (m ⊗ 1 λ ). Thus it maps an element 1 ⊗ 1 ⊗ r to x ⊗ 1 ⊗ r + 1 ⊗ x ⊗ r.
The case for |j − i| > 1 is easy because the bimodules for E i E j and E j E i are equal.
There are four non-trivial case for (λ i , λ i+1 , λ i+2 ). Let a and b be crossingless matches. Let B be the bimodule for E i E i+1 and let B ′ be the bimodule for E i+1 E i .
• Suppose the circle passing through point i on the bottom row of the tangle for E i E i+1 is different from the circle passing through point i + 1 on the bottom row. Then a B b = A ⊗ A ⊗ R and a B ′ b = A ⊗ R. Then ψ i,i+1 = m ⊗ 1. Then it is easy to verify that ψ i,i+1 Case 2: (λ i , λ i+1 , λ i+2 ) = (0, 1, 1). Similar to case 1.
• Suppose the circle passing through point i on the bottom row of the tangle is the same as the circle passing through point i + 1 on the bottom row. Then a B b = A ⊗ R and a B ′ b = A ⊗ A ⊗ R. Then ψ i,i+1 is given by ∆ ⊗ 1. This then follows as in case 1.
Proof. The proof of the first part consists of verifying the equality in many different cases, each of which is similar to the second part. We only prove the second part in the case j = i + 1 as the case j = i − 1 is similar. There are four cases for ( Case 1: (λ i , λ i+1 , λ i+2 ) = (0, 1, 1). In this case, (ψ j,i 1 i ) • (1 j ψ i,i ) • (ψ i,j 1 i ) = 0 because it passes through the functor E i+1 E i E i which is zero on the category corresponding to this λ. On the other hand Let B be the bimodule for the functor E i E i+1 E i . Then this is a sequence of maps where the first map given by comultiplication, the middle map is given by the map 1 ⊗ κ, and the last map is multiplication. This sequence of maps acts on 1 ⊗ α ∈ B as follows: Case 2: (λ i , λ i+1 , λ i+2 ) = (0, 2, 2). This is similar to case 1 except that now ( Case 3: (λ i , λ i+1 , λ i+2 ) = (0, 1, 2). In this case, (ψ j,i 1 i ) • (1 j ψ i,i ) • (ψ i,j 1 i ) = 0 since this map passes through the functor E i+1 E i E i which is zero on the category corresponding to λ.
On the other hand Let B be the bimodule for the functor E i E i+1 E i . Then this is a sequence of maps where the first and third maps are given by lemmas 2 and 3 respectively, and the middle map is given in section 4.5. This sequence of maps acts on 1 ⊗ α, x ⊗ α ∈ B as follows: Case 4: (λ i , λ i+1 , λ i+2 ) = (0, 2, 1). This is similar to case 1 except that now (1 i Theorem 1. There is a 2-functor Ω k,n : KL → HK k,n such that for all i, j ∈ I, Ω k,n (I λ ) = I λ , Ω k,n (Y i;λ ) = y i;λ , Ω k,n ( i;λ ) = ∪ i;λ , For i = 1, . . . , 2k, let e ij denote the (i, j)-matrix unit, and let ε i ∈ d * be the coordinate functional ε i (e jj ) = δ ij . Let O be the category of finitely generated g-modules which are diagonalizable with respect to d and locally finite with respect to p. Let denote the weight lattice and root lattice of gl 2k , respectively. The dominant weights are given by the set Let µ and µ ′ be integral dominant weights of g, and let Stab(µ) denote the stabilizer of µ under the ρ-shifted action of the symmetric group S 2k . Suppose µ ′ − µ is an integral dominant weight. Then, let θ µ ′ µ : O be the translation functor of tensoring with the finite dimensional irreducible representation of highest weight µ ′ − µ composed with projecting onto the µ ′ -block, and let θ µ µ ′ be its adjoint. Let P µ be a minimal projective generator of O µ . It was shown that A µ = End g (P µ ) has the structure of a graded algebra [1]. Since O µ is Morita equivalent to A µ -mod, we consider the category of graded A µ -modules which we denote by Z O µ . Let the graded lift of O (k,k) µ and P (k,k) µ be Z O (k,k) µ and Z P (k,k) µ , respectively. It is known that if Stab(µ) ⊂ Stab(µ ′ ), there is a graded lift of the translation functors, cf. [18], which by abuse of notation we denote again by θ µ µ ′ and θ µ ′ µ .
The key tool in the construction of graded category O is the Soergel functor. Let λ = (λ 1 , . . . , λ n ) be a composition of 2k, let S λ = S λ1 × · · · × S λn , let w µ 0 be the longest coset representative in S 2k /S µ , and let P (w µ 0 · µ) be the unique up to isomorphism, indecomposable projective-injective object of O µ . Let C = S(h)/S(h) S 2k + be the coinvariant algebra of the symmetric algebra for the Cartan subalgebra with respect to the action of the symmetric group. Let x 1 , . . . , x 2k be a basis of S(h) and by abuse of notation also let x i denote its image in C. Let C λ be the subalgebra of elements invariant under the action of S λ . Soergel proved in [16]: Define the Soergel functor V µ : O µ → C Stab(µ) -mod to be Hom g (P (w 0 .µ), •).
Proposition 19. Let P be a projective object. Then there is a natural isomorphism Proof. This is the Structure Theorem of [16].
The set of objects of P k,n are the categories Z P (k,k) λ , λ ∈ P (V 2ω k ).
For each i ∈ I, we define functors E i I λ , and K i I λ To this end, let λ be a weight of V 2ω k and i ∈ I + . Then we have compositions of 2k into n + 1 parts: Also, if λ = i a i ω i ∈ P , set r i,λ = 1 + a 1 + · · · + a i−1 + a i+1 and s i,λ = 2 − a i − a i+1 .
There is also an isomorphism of algebras: where J λ(−i),n is the ideal generated by the homogeneous terms in the equation  x(λ) i,r t r λj s=0 x(λ) j,s t s = 1.
Proof. This now follows from the computations in [11, Section 6.2] for bimodules over the cohomology of flag varieties using the naturality of the isomorphism in proposition 19.
Finally we show that the category P k,n is a categorification of the module V 2ω k . Denote the Grothendieck group of P k,n by [P k,n ], and let [P k,n ] Q(q) = C(q) ⊗ Z[q,q −1 ] [P k,n ].

Proposition 21.
There is an isomorphism of U q (sl n ) modules [P k,n ] Q(q) ∼ = V 2ω k .
Proof. Since projective functors map projective-injective modules to projective-injective modules, it follows from Theorem 2 and [11], that [P k,n ] Q(q) is a U q (sl n )-module. By construction, it contains a highest weight vector of weight 2ω k so it suffices to compute the dimension of its weight spaces.
By [3,Theorem 4.8], the number of projective-injective objects in O Let S = {i ∈ I + |λ i = 1}. Denote by |S| the cardinality of this set. Consider a Young diagram with |S| 2 rows and 2 columns. Let T ′ denote the set of tableau on such a column with entries from S such that the rows and columns are decreasing. It is well known that the cardinality of the set T ′ is the Catalan number ( 2|S| |S| ) |S|+1 . There is a bijection between T and T ′ . For any tableaux t ′ ∈ T ′ one constructs a tableaux t ∈ T by inserting a new box with the entry i in each column for each i ∈ I + such that λ i = 2. The inverse is given by box removal.
Finally, the Weyl character formula gives that the dimension of the λ weight space of V 2ω k is ( 2|S| |S| ) |S|+1 .